![14 1. Constructive Type Theory: Foundation and Formalization
expanding the system by applying functions to functions: in the Grundge-
setze der Arithmetik, Frege established that the formula (α1, . . . , αn) is the
“type” (using the modern terminology) of n-place functions, which gives
rise to an object of a specific level (Stufe).19
This means that if α1 up to
αn are types, we can form a new type which collects all the previous ones
(here we already introduce a formalization for such a predication):
α1 : type, . . . , αn : type
(α1, . . . , αn) : type
A schema of the correspondence with functions is the following:
() object (Gegenstand)
(()) unary function
(()()) binary function
(() . . . ()n) n-ary functions
which take many functions into an object (Wertverlauf )
where clearly unary functions have objects as arguments, secondary func-
tions have unary functions as arguments, and so on. Later, in his Begriff-
schrift, Frege introduced the judgeable contents (beurteilbare Inhalte), con-
sidering propositions inside the universe of objects: this gave rise to anti-
nomies due to impredicativity. Russell (1903) presents a way out from the
paradoxes generated by Frege’s functional hierarchy, and in this sense it
represents the natural ancestor of Type Theory. The Russellian type theory
is related to the Fregean functional hierarchy by accounting the “simple”
types, independently from the complexity of definition (so that it naturally
reflects the order of “objects”, “concepts”, “second order concepts”, etc.).20
Russell presented the simple theory of types in two appendices (1903), then
developed the ramified version (1908): in this new version the type of a
function depends not only on the types of its arguments, but also on the
types of entities referred to, and quantified over, by the function itself,
i.e. through typing propositions. In the simple theory of types Russell mod-
ified the Fregean structure by defining:
– The type of the individual valued functionals
– The type of proposition-valued functions
Referring again to the previous case, when α1, . . . , αn are types, the
Russellian theory understands [α1, . . . , αn] as a type too, i.e. n-ary propo-
sitional functions with types α1, . . . , αn as arguments represent types
themselves:
α1 : type, . . . , αn : type
[α1, . . . , αn] : type
19
The level of a type is defined by Frege (1884) as the maximum of the levels of
the argument types plus one.
20
One should also remember the anticipation of the simple theory of type due to
Schröder. Cf. Church (1976).](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-29-320.jpg)
![1.3. Types 15
and the system is then enlarged adding clauses for relational types:
[] = type e.g ⊥,
[[]] = type(type) unary function, e.g ¬
[[], []] = type((type), (type)) binary function, e.g. ∧, ∨, ⊃
[[[]]] = type(type(type)) quantified function, e.g. ∀, ∃
[[] . . . []n] = type(type) . . . (type)n type of n-ary functions/relations
Both structures, the Fregean and the Russellian, are powerful enough to
express systems of a certain complexity, such as in the context of first-order
logic, but not enough for more complex systems. A new notation for the
theory of types was then introduced by Schönfinkel in 1924, based on the
idea of representing functions of n arguments as a unary function having a
value corresponding to a function with n − 1 arguments, proceeding until
one reaches the ground types (individuals and propositions). In this way, it
is possible to give three clauses for forming types:
1. ι (for individuals) is a type.
2. o (for propositions) is a type.
3. If α and β are types, then (βα) is a type (with α for the argument type
and β for the value type).
Accordingly the level is defined in the following way:
L(ι) = L(o) = 0 L((αβ) = max(L(α) + 1, L(β)),
which represents the basic structure for Church’s notation, and for the
structures developed by Schütte, Curry, and Ajdukiewicz.21
In general, the
simple type structure makes it possible to type all the constants of first-
order logic, while with dependent types of Intuitionistic type theory one is
able to type even quantifiers whose domains vary.
According to Martin-Löf something is never an entity without being of
a certain sort or kind, and each mathematical object is always typed: such
types (as we will see later) are the source of the categories of predication,
giving rise to the syntax and semantics of the theory. Whenever the notion
of type is understood in this deep and broad philosophical aspect, being
assimilated in a general and intuitive sense to a structure constituting a
family of objects determined by any property, together with an equivalence
relation, the resulting formal theory is of a specific kind: such a notion
of type is conceptually prior to, and provides an interpretation for, other
notions such as the one of proposition, or the mathematical ones of set,
elements of a set, the set-valued functions over a given set, and predicates
over a given set. Thus, a theory of types can be used to present a theory of
sets, using variables ranging over sets and higher-order objects, but in fact
by choosing to use the more general and basic interpretation of the notion
21
Martin-Löf (1993) has treated the modern evolution of the notion of type.](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-30-320.jpg)
![28 1. Constructive Type Theory: Foundation and Formalization
i.e. their meanings.45
The first kind of identity is given by the sign “=def ”,
so that when one refers to a relationship such as
a =def b
it concerns the identity between two expressions. The second type of relation
holds between the objects one is talking about, and it is formalized as
a = b.
The informal reasoning behind the interderivability of these two kinds of
formulae is the following: given the identity axiom a = a, the formula a =def
b implies that (a = a) =def (a = b), so that a = a and a = b have the same
meaning and this immediately gives us the conclusion a = b.46
On the other
hand, the identity of the objects a and b is enough to state the identity of
the respective expressions, so that it holds the formula a =def b. Moreover,
on the basis of the Intuitionistic approach, the validity of the judgement
a = b must of course correspond to the possession of its derivation. If such
a closed derivation is supposed to hold, then it is clear that the identity
a = b implies the interconvertibility (formal counterpart of the informal
“definitional equality”) of the terms a and b. In this sense we can say that
two derivations are interconvertible if, and only if, the proofs that they
represent are identical, so that “identical” means in this context “provably
identical”. The relation of definitional identity is expressed by Martin-Löf
according to three principles,47
each of them having a formal counterpart,
namely, a conversion rule:
1. Definitional equality between the definiens and its definiendum:
redex conv convertum(...)
2. Substitution of definitionally equal expressions for a variable in a given
expression leads to definitionally equal expressions (preservation of defi-
nitional equality under substitution):
a conv c
b[a] conv b[c]
3. Definitional equality is reflexive, symmetric, transitive:
a conv a
a conv b
b conv a
a conv b b conv c
a conv c
These are the general formal rules holding for the logical notion of identity
and instantiated in the forms of judgements. Such rules allow the formula-
tion of judgements either for a set-theoretical or for a propositional system,
45
Ibid. (1975a, pp. 101–104).
46
Ibid. (1975a, p. 102).
47
Ibid. (1975a, p. 93).](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-43-320.jpg)
![1.4. Identity 29
via the expression of definitional identity between elements of a set and
proof objects for propositions. In this way one understands the difference
between definitional or semantical identity (a = b) and syntactically induced
identity (a =def b): the first one is much stronger than the second one, in-
volving the ontological aspect of the theory; syntactical identity refers only
to the identity of the formal expressions we use to express objects.48
1.4.2 Identity as Theoretical Notion
The identity issue in its philosophical and logical aspect has been widely
considered since antiquity. It is clearly already present in Platonic dialogues,
entering in the explanation of the relation of things to ideas; Aristotle brings
this problem to its core, by considering the nature of essence, for which the
notion of “being” is treated in connection to “predication”, thus referring
to the categories of “sameness”, “otherness”, and “contrariety”.49
Identity
is explained in terms of predication when Aristotle says that two things
are identical if all that is predicated (or predicable) of one of them, is
predicable of the other.50
The principle of identity (“a being is what it is”)
is thus obviously central to the Aristotelian philosophy, and it notoriously
expresses the positive formulation of the basic principle of contradiction:
“a being cannot both be and not be at the same time and under the same
respect”,51
the logical principle par excellence, both a principle of knowledge
and reality. Since these first formulations, the identity issue and the relation
to definition was a central topic in Scholastics, particularly in Ockham, with
the distinction between definitio quid rei and definitio quid nominis.52
But
one has to wait until Frege to find a fruitful connection for the development
of the notion of identity with that of type: by using a unique universal type
(the sort of all objects), Frege explained the question of identity both in
terms of equality of content (Inhaltgleichheit) and as the binary relation of
identity, respectively in the Begriffschrift and in the Grundgesetze; in Über
Sinn und Bedeutung he explains the identity “=” as a relation between
48
Note that within the Hilbertian formalistic perspective, the semantical identity
criterion can be only reduced to this syntactic criterion, because all expressions
are empty of their meaning. In Sections 1.5.4 and 1.5.5 the ground types for sets
(set) and propositions (prop) will be introduced; the notion of identity presents
the following meanings within those types:
– With A = B : prop the concept of material equivalence between two proposi-
tions is intended.
– With A = B : set the concept of equipotency between two sets is intended.
49
Martin-Löf refers in particular to Metaphysics, book ∆, 1018a35-39; cf. Martin-
Löf, (1993, pp. 41–42).
50
Aristotle [Top] book 7, cap. 1, 152a 5–30.
51
Cf. Aristotle [Metaph], book Γ, 1006a 3–5: “ἡμεῖς δὲ νῦν εἰλήφαμεν ὡς ἀδυνάτου
ὄντος ἃμα εἶναι καὶ μὴ εἶναι.”
52
Ockham (1324), Logica terminorum, III, 26.](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-44-320.jpg)
![1.5. Formal Analysis of Types and Judgements 31
1.5 Formal Analysis of Types and Judgements
In Section 1.5.1 the ground types, namely, the type of sets and that of
propositions, will be introduced as objects definable by monomorphic types;
the forms of judgements in use within the theory and their formalization
will be shown. Moreover, this will also allow for the introduction of another
type, that of functions and its formalization. On the basis of the theoretical
analysis developed above, the formal role of the notion of identity will be
explained and formal rules for it introduced.
1.5.1 Formalizing the Forms of Judgement
The aim of this section is to introduce the formalization of the judgemental
forms used in the type-theoretical framework. It will become clear why
the notion of function and the notations explained above for this concept
(those developed by Frege, Russell, and later by Church) are relevant to the
notion of type considered here. I will first consider the formal expressions of
the theory and try to explain how the formalization reflects the theoretical
questions introduced above: conceptual priority, identity, and definition.
The essential questions for defining the formal objects of the theory are:
What is a type? And what does it mean for an element to be of a certain
type? According to the definition of type, and the explanation of mathe-
matical objects already considered in the light of Intuitionistic logic, it is
not possible to know what a type is and being at the same time in doubt
about the properties of the objects belonging to that type: this obviously
makes the theory more trustworthy from an epistemic point of view. There
are then two complementary ways to follow, in order to get the definition
of “type”:
(a) To know a type is to know what an object of that type is.
(b) To know a type means knowing what it means for two objects of that
type to be the same.
Accordingly, two main forms of judgements are obtained: the first will state
that there is an element belonging to a certain type (a), the second that
two objects are identical within the same type (b). These expressions will
together furnish the definition for the type involved. According to the con-
ceptual priority explained in Section 1.3.3, the type denoted by such a
definition must be meaningfully stated before, i.e. its meaningfulness is a
presupposition for those judgements to be done: in the conceptual order,
the type comes before its definition.59
Let us start by the formalization of
type-expressions. The basic judgement
α is a type
59
Cf. Martin-Löf (1993, pp. 31–32). The problem of priority between type-
declarations and type-definitions is just introduced here: the classic solution pro-
vided by Martin-Löf in order to avoid impredicativity is presented in Section 1.5.3.
A new critical treatment of the problem and a theoretical solution is presented
throughout Chapter 4. For the formal presentation of presuppositions in CTT see
also Primiero [forthcoming a].](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-46-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
United States, you will have to check the laws of the country where
you are located before using this eBook.
Title: Zones of the Spirit: A Book of Thoughts
Author: August Strindberg
Commentator: Arthur Babillotte
Translator: Claud Field
Release date: November 6, 2013 [eBook #44118]
Most recently updated: October 23, 2024
Language: English
Credits: Produced by Marc D'Hooghe (Images generously made
available by the Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK ZONES OF THE
SPIRIT: A BOOK OF THOUGHTS ***](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-60-320.jpg)
!['Providence is planning something with thee, and this is the
beginning of thy education.'[1]
Soon after this he gave up his chemical experiments and took up
alchemy, with a conviction, almost pathetic in its intensity, that he
would succeed in making gold. Although his dramas had already
been performed in Paris, a success which had fallen to the lot of no
other Swedish dramatist, he forgot all his successes as an author,
and devoted himself solely to this new pursuit, to meet again with
disappointment.
On March 29, 1897, he began the study of Swedenborg, the
Northern Seer. A feeling of home-sickness after heaven laid hold of
him, and he began to believe that he was being prepared for a
higher existence. I despise the earth, he writes, this unclean
world, these men and their works. I seem to myself a righteous
man, like Job, whom the Eternal is putting to the test, and whom the
purgatorial fires of this world will soon make worthy of a speedy
deliverance.
More and more he seemed to approach Catholicism. One day he, the
former socialist and atheist, bought a rosary. It is pretty, he said,
and the evil spirits fear the cross. At the same time, it must be
confessed that this transition to the Christian point of view did not
subdue his egotism and independence of character. It is my duty,
he said, to fight for the maintenance of my ego against all
influences which a sect or party, from love of proselytising, might
bring to bear upon it. The conscience, which the grace of my Divine
protector has given me, tells me that. And then comes a sentence
full of joy and sorrow alike, which seems to obliterate his whole past.
Born with a home-sick longing after heaven, as a child I wept over
the squalor of existence and felt myself strange and homeless
among men. From childhood upwards I have looked for God and
found the Devil. He becomes actually humble, and recognises that
God, on account of his pride, his conceit, his ὕβρις, had sent him for
a time to hell. Happy is he whom God punishes.](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-65-320.jpg)
![the canticle of his life, a hymn of thankfulness for the recovered faith
in which he has found peace. At its conclusion he thus sums up:
Rousseau's early doctrine regarding the curse of mere learning
should be repondered.
A new Descartes should arise and teach the men to doubt the
untruths of the sciences.
Another Kant should write a new Critique of Pure Reason and re-
establish the doctrine of the Categorical Imperative, which, however,
is already to be found in the Ten Commandments and the Gospels.
A prophet should be born to teach men the simple meaning of life
in a few words. It has already been so well summed up: 'Fear God,
and keep His commandments,' or 'Pray and work.'
All the errors and mistakes which we have made should serve to
instil into us a lively hatred of evil, and to impart a fresh impulse to
good; these we can take with us to the other side, where they will
bloom and bear fruit. That is the true meaning of life, at which the
obstinate and impenitent cavil, in order to save themselves trouble.
Pray, but work; suffer, but hope; keeping both the earth and the
stars in view. Do not try and settle permanently, for it is a place of
pilgrimage; not a home, but a halting-place. Seek the truth, for it is
to be found, but only in one place, with the One who Himself is the
Way, the Truth, and the Life.
ARTHUR BABILLOTTE.
[1] Strindberg's Inferno.
CONTENTS
THE HISTORY OF THE BLUE BOOK
A BLUE BOOK—](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-68-320.jpg)
![wish to write cheerfully and beautifully, but ought not, and cannot. I
conceive it as a terrible duty to be truthful, and life is indescribably
hideous.
Now the clock strikes eleven, and at twelve o'clock is the rehearsal.
The same day at 8 P.M. I have seen the rehearsal of the Dream Play,
and suffered greatly. I received the impression that this piece ought
not to be played. It is presumptuous, and certainly blasphemous (?).
I am disturbed and alarmed.
I have had no midday meal; at seven o'clock I ate some cold food
out of the basket in the kitchen.
During the religious broodings of my last forty days I read the Book
of Job, saying to myself certainly at the same time that I was no
righteous man like him. Then I came to the 22nd chapter, in which
Eliphaz the Temanite unmasks Job: Thou hast taken pledges of thy
brother for nought, and stripped the naked of their clothing; thou
hast not given water to the weary to drink, and thou hast withholden
bread from the hungry. ... Is not thy wickedness great and thine
iniquities infinite?
Then the whole comfort of the Book of Job vanished, and I stood
again forlorn and irresolute. What shall a poor man hold on to? What
shall I believe? How can he help thinking perversely?
Yesterday I read Plato's Timæus and Phædo. There I found so much
self-contradictory wisdom, that in the evening I threw my devotional
books away and prayed to God out of a full heart. What will happen
now? God help me! Amen.
The stage-manager visited me yesterday evening. We both felt, in
despair.... The night was quiet.
April 16, 1907.—Read the proof of the Black Flags,[1] which I wrote
in 1904. I asked myself whether the book was a crime, and whether
it ought to be published. I opened the Bible, and came on the
prophet Jonah, who was compelled to prophesy although he hid
himself. That quieted me. But it is a terrible book!](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-78-320.jpg)
![Like a swallow or a crane, so did I chatter; I did mourn as a dove:
mine eyes fail with looking upward.
Lord, I am oppressed; undertake for me.
What shall I say? He hath both spoken unto me, and himself hath
done it.
Behold, it was for my peace that I had great bitterness;
Thou hast in love to my soul delivered it from the pit of corruption.
The living, the living, he shall praise thee, as I do this day.
The father to the children shall make known thy truth.
I saw beforehand what awaited me if I broke with the Black Flags.
But I placed my soul in God's hands, and went forwards. I affix as a
motto to the following book, He who departeth from evil, maketh
himself a prey.
The strangest thing, however, is that from this moment my own
Karma began to complete itself. I was protected, things went well
with me, I found better friends than those I had lost. Now I am
inclined to ascribe all my former mischances to the fact that I served
the Black Flags. There was no blessing with them!
[1] A roman à clef in which Strindberg fiercely attacks the
Bohemians and emancipated women of Stockholm.
A BLUE BOOK
The Thirteenth Axiom.—Euclid's twelfth axiom, as is well known,
runs thus: When one straight line cuts two other straight lines so](https://image.slidesharecdn.com/4985975-250526112843-8401f7f6/85/Information-And-Knowledge-A-Constructive-Typetheoretical-Approach-Primiero-81-320.jpg)
Information And Knowledge A Constructive Typetheoretical Approach Primiero
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- 7. LOGIC, EPISTEMOLOGY, AND THE UNITY OF SCIENCE Editors Shahid Rahman, University of Lille III, France John Symons, University of Texas at El Paso, U.S.A. Editorial Board Jean Paul van Bendegem, Free University of Brussels, Belgium Johan van Benthem, University of Amsterdam, the Netherlands Jacques Dubucs, University of Paris I-Sorbonne, France Anne Fagot-Largeault Collège de France, France Bas van Fraassen, Princeton University, U.S.A. Dov Gabbay, King’s College London, U.K. Jaakko Hintikka, Boston University, U.S.A. Karel Lambert, University of California, Irvine, U.S.A. Graham Priest, University of Melbourne, Australia Gabriel Sandu, University of Helsinki, Finland Heinrich Wansing, Technical University Dresden, Germany Timothy Williamson, Oxford University, U.K. Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light no basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity. VOLUME 10
- 8. By Information and Knowledge Approach Giuseppe Primiero Ghent University, Belgium A Constructive Type-theoretical
- 9. A C.I.P. Catalogue record for this book is available from the Library of Congress. P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper Cover image: Adaptation of a Persian astrolabe (brass, 1712–13), from the collection of the Museum of the History of Science, Oxford. Reproduced by permission. All Rights Reserved ISBN 978-1-4020-6169-1 (HB) Published by Springer, ISBN 978-1-4020-6170-7 (e-book) No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, use by the purchaser of the work. and executed on a computer system, for exclusive specifically for the purpose of being entered without written permission from the Publisher, with the exception of any material supplied c ° 2008 Springer Science + Business Media B.V.
- 10. Logic, which alone can give certainty, is the instrument of proof; intuition is the instrument of invention. H. Poincaré, La valeur de la Science . . . and he knows absolutely—knows it all the way, deep as knowing goes, he feels the knowledge start to hammer in his runner’s heart— that he is uncatchable. D. De Lillo, Underworld Information is not knowledge. A. Einstein
- 12. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Constructive Type Theory: Foundation and Formalization . . . . . . . . . . . . . . . . . . . . 7 1.1 Philosophical Foundation . . . . . . . . . . . . . . . . . . . 7 1.2 Basic Epistemic Notions . . . . . . . . . . . . . . . . . . . . 8 1.3 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Constructive Notion of Type . . . . . . . . . . . . . 11 1.3.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.3 Conceptual Priority . . . . . . . . . . . . . . . . . . 18 1.3.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4.1 Definitional Identity vs. Syntactically Induced Identity . . . . . . . . . . . . . . . . . . . . 27 1.4.2 Identity as Theoretical Notion . . . . . . . . . . . . 29 1.5 Formal Analysis of Types and Judgements . . . . . . . . . . 31 1.5.1 Formalizing the Forms of Judgement . . . . . . . . . 31 1.5.2 Formalizing Equality Rules . . . . . . . . . . . . . . 33 1.5.3 Categories . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.4 Type set . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5.5 Type prop . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.6 The Notion of Proposition for CTT . . . . . . . . . 40 1.5.7 Propositions as Sets . . . . . . . . . . . . . . . . . . 46 1.6 Dependent Objects: Hypothetical Judgements . . . . . . . . 47 1.6.1 Judgements Depending on One Assumption . . . . . 50 1.6.2 Judgements Depending on More Assumptions . . . . 51 1.7 Introducing Functions . . . . . . . . . . . . . . . . . . . . . 52 1.8 Computational Rules . . . . . . . . . . . . . . . . . . . . . . 54 1.8.1 The System of Rules and Some Examples for set and prop . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.8.2 Rules for the type Func . . . . . . . . . . . . . . . . 57 1.9 Introducing Information . . . . . . . . . . . . . . . . . . . . 60 vii
- 13. viii Contents 2 Analyticity and Information . . . . . . . . . . . . . . . . . . . . . 63 2.1 At the Origin of the Problem . . . . . . . . . . . . . . . . . 64 2.1.1 The Modern Origin of Analyticity: Kant . . . . . . . 64 2.1.2 Elements of the Bolzanian Doctrine of Science . . . 70 2.1.3 A New Concept of Analyticity (Against the Critical View) . . . . . . . . . . . . . . 73 2.1.4 Analyticity in Question: The Possibility of Knowledge . . . . . . . . . . . . . . . . . . . . . . 77 2.2 Analysis and Synthesis . . . . . . . . . . . . . . . . . . . . . 84 2.2.1 Act and Content: A Foundational Distinction . . . . 87 2.2.2 Content and Meaning . . . . . . . . . . . . . . . . . 88 2.2.3 Analyticity Reconsidered: From Meaning to Information . . . . . . . . . . . . . . . . . . . . . 93 2.2.4 Rejecting the Analytic/Synthetic Distinction: Quine 95 2.2.5 Towards a Constructive Notion of Analyticity . . . . 99 2.3 Informativeness of Derivations . . . . . . . . . . . . . . . . . 101 2.3.1 Individuals and Degrees: Computing Information of Sentences . . . . . . . . . . . . . . . . . . . . . . . 103 2.4 Different Notions of Information . . . . . . . . . . . . . . . 109 2.4.1 Conceptual vs. Contentual Information . . . . . . . 111 2.4.2 Surface Information: Probability and Possible Worlds 111 2.4.3 Increasing Logical Information: Depth Information . 113 2.5 Basic Elements of a Knowledge System . . . . . . . . . . . . 115 2.5.1 Reconsidering the Semantic Approach . . . . . . . . 115 2.5.2 Recollecting Perspectives on Information . . . . . . . 119 2.5.3 Knowledge: What, That, How . . . . . . . . . . . . . 120 3 Formal Representation of the Notion of Information . . . . . . . 125 3.1 CTT as the General Framework: Informal Description . . . 125 3.1.1 Formalization of Knowledge and Information . . . . 129 3.1.2 Contexts: Formal Explanation . . . . . . . . . . . . 130 3.2 Representation of Knowledge and Information . . . . . . . . 135 3.2.1 Presuppositions . . . . . . . . . . . . . . . . . . . . . 136 3.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . 139 3.2.3 Types and Meaning Declarations . . . . . . . . . . . 141 3.2.4 Truth and the Role of Assumptions . . . . . . . . . 144 3.2.5 Defining Information . . . . . . . . . . . . . . . . . . 148 3.3 Contexts as Constructive Possible Worlds . . . . . . . . . . 150 3.3.1 Introducing Orderings: Kripke Models . . . . . . . . 152 3.4 The Knowledge Framework . . . . . . . . . . . . . . . . . . 154 3.4.1 Updating Information, Extending Knowledge . . . . 154 3.4.2 The Structure of Knowledge . . . . . . . . . . . . . . 158 4 Constructive Philosophy of Information . . . . . . . . . . . . . . 165 4.1 An Extension for the Constructive Epistemology . . . . . . 165
- 14. Contents ix 4.2 Information and Mathematics . . . . . . . . . . . . . . . . . 168 4.2.1 From Analytic Method to the Analyticity of Logic . 170 4.3 The Role of Constructions . . . . . . . . . . . . . . . . . . . 171 4.4 Types and Categories of Information . . . . . . . . . . . . . 175 4.4.1 Overview on Presuppositions Theory and Dynamic Logics . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.4.2 Declaring and Explaining Meanings . . . . . . . . . 179 4.4.3 Meaning and Predication . . . . . . . . . . . . . . . 184 4.5 Information and Logical Knowledge . . . . . . . . . . . . . 188 4.6 Final Epistemic Foundation for Information . . . . . . . . . 191 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
- 16. Introduction This research is the result of a fruitful connection and provides a signifi- cant link between two topics of a logical and philosophical enquiry. It tries to provide a solution to the problem of analyticity: with this expression I understand, on the one hand, the essential nature of analytic truths and, on the other, the related explanation of the analytic nature of logical inference. The connection between these two sides of what will be referred to as the Analyticity Principle, can be briefly explained as follows: by analytic truth one understands in general a sentence whose content is logically true; by logically true one understands moreover truth independent from matters of fact or empirical data, a truth which is therefore established by logical criteria only. On this basis, it follows that a logical inference represents a purely analytic process, in opposition to its property of being able to produce knowledge, a situation which is exemplified by the conflicting no- tions of validity and utility. The question-begging topic of this research is therefore that of analyticity, the inspiring problem for which a solution is formulated in the present book. If analyticity represents the starting point of this research, the other part of its content is the result of a far more complex question; to represent the notion of Information in the context of logical calculi. The main idea of this research can therefore be formulated in the following terms: to find an intuitive and formally useful representation of the notion of information within a logical setting, in order to provide a clear formulation of the analyticity principle. The logical formulation is provided by the constructive version of Type Theory. This research is thus part of a precise field of currently growing perspectives and theories, only recently explicitly recognized under the label of Philosophy of Information: by this term one refers to the criti- cal investigation on the conceptual nature and on the basic principles of information, the determination of the relevant computational systems for such a notion, and the description of its use; it moreover expresses the philosophical formulation of problems related both to epistemology and technology. Therefore, the Philosophy of Information collects a wide range of philosophical investigations. Concerning the present research, the problem of analyticity represents the essential topic in the connection between logic and information. Information will be thus referred to as the 1
- 17. 2 Introduction conceptual term expressing the content of logical derivations: to provide a proper interpretation of such a content in a precise formal meaning is a first result of this research. The notion of information is in general de- fined according to either a result-based approach or rather an agent-based one: this obviously depends on the kind of philosophical conditions one is willing to satisfy or to consider relevant. The present interpretation is strongly influenced by the logical framework accepted, and thus it provides an all-invasive reformulation of the principles usually assumed to hold in the context of the various theories of information, in particular regarding the alethic value ascribed to such a notion. This depends on the description of the logical approach used, and of the results considered relevant to the proposed solution. The logical and philosophical perspective accepted throughout this re- search is thus essential to the understanding of the notions involved, to the reformulation of the concept of information, and to the proposed solution to the problem of analyticity. The understanding of a logician’s attitude to- wards truth and knowledge is at the very basis of an entire train of thoughts and of the choices about what reality and truth are, what it means to know, and obviously the consequences one is willing to accept from this. The con- structive approach represents in this sense a way of accepting responsability for our own decisions, determining knowledge in terms of our own limits, and possibly establishing a dependence of our reality on the steps one chooses (or is able) to make, an approach which reflects also an ethics of knowledge. In this light, the constructive perspective provides an important and essen- tial change: explaining information does not amount just to understanding what is expressed by a certain propositional content; rather, it is clarified also in terms of what is needed in order for a certain judgemental content to be formulated. Formally, this leads to ascribe a relevant role to the notion of assertion condition. Moreover, the resulting notion is developed in line with the logical elements and concepts furnished by the formalization; thus, it relays on a solid logical analysis. The process of verifying an intuition may have more or less fruitful re- sults, and it can even be wrong. To my mind this is exactly the role played by the formalism, to justify and prove whatever one feels could be the right model or the correct formulation of an idea. This process leads in the present case to Constructive Type Theory (CTT) as the framework which actually provides not only the formal but also the philosophical theory: the theory developed by Martin-Löf is in my opinion philosophically powerful and provides a high degree of conceptual awareness. The ability of devel- oping a deep conceptual framework is essential to the work of logicians coming from philosophy, whereas mathematicians and computer scientists would value other properties in a theory. In this direction the role of the present research is twofold, showing a formal development for a certain the- ory and suggesting a theoretical extension of the epistemic analysis at the basis of the relevant philosophical logic. The formal development of this
- 18. Introduction 3 theory brings to recognize a deep and essential change in the epistemic background: the suggested extension of the constructive epistemology via the notion of information represents a step towards the perfect matching between the constructive philosophy and its formal logic, a second result of this work. Chapter 1 introduces the foundation and formalization of CTT as the working framework: the theory is presented in its formal setting, but it mainly provides a new analysis of its philosophical themes. In particular, one will find some philosophical topics which are hardly considered in other introductions, whereas the technical and mathematical structure of CTT is well known and continuously developed. My aim here is to explore the theoretical possibilities of the theory, making reference to ancestors of the solutions proposed within CTT, directing attention to the epistemology and to the formal objects introduced for the problem at hand. In particular, the introduction of the category/type distinction and the explanation of the calculus of contexts is essential. CTT proposes a proper ontology, reflected in a hierarchical structure of types, forming in this way both the linguis- tic and the objectual levels of the structure we are speaking about. The hierarchical structure of types (and of their elements) can be thought of as a system (database) of informations available within the theory; in this sense, CTT reflects perfectly the notion of “ontology” as intended within computer science. The chapter ends by introducing the usually intended notion of information in the context of CTT: in this sense, CTT is a system which fully treats information, i.e. it is procedurally analytic, and it gives the ability of forgetting and recovering information in terms of an abstrac- tion procedure. Nevertheless, this still refers to information only in a purely computational sense, whereas my aim here is to introduce an epistemic and formal description of that notion: this aim is obtained in the constructive setting by defining the essential difference from the concept of knowledge. The resulting notion is user-dependent and epistemically defined, it avoids the difficulties coming from the alethic nature imposed by the realistic ap- proach and it presents an interesting and strong connection to meaning theory. The basis of such a formulation is contained in the strong commit- ment the constructivist owes to a notion of truth defined as existence of a proof: this in fact implies a stronger obligation in what he/she is disposed to accept, and eventually what he/she can later dismiss. In Chapter 2, I shall present the problem (analyticity) and introduce the development of its possible solutions, up to the introduction of the no- tion of information. The analyticity principle is developed starting with the approaches of two great philosophers, Kant and Bolzano, in order to un- derline that the dichotomy between act and content (a central topic of the constructive approach in logic, in terms of the distinction between act of judgement and propositional content) is a natural theoretical consequence at the basis of the definitions of analyticity and analytic truth given by the two authors. Both conceptions aim at a description of scientific processes:
- 19. 4 Introduction Kant connects analyticity to total uninformativeness of deductive processes, whereas Bolzano goes the other way round, using this notion to general- ize the concept of validity in order to define derivability. The two authors characterize analyticity by the conceptual shift determined in the different definitions: thus, on the basis of the mentioned distinction between act and content, first the notion of meaning is introduced (mainly by referring to the work of Frege and Ayer) and finally the definition of analyticity is pre- sented in terms of the notion of information (which historically is due to Hintikka, who takes over some Kantian insights). Chapter 2 finally leads to the mentioned epistemic description of the notion of information, based on a constructive reformulation of some basic principles: this means also to provide the conceptual lines along which a formal description of epistemic information within knowledge processes can proceed. Chapter 3 introduces the formal structure which expresses the notion of information within CTT. Such a formalization does not present another framework to organize information within databases (one of the most basic applications provided by Informational Logics); it does not just draw a logi- cal framework for some specific semantic approach to information. It rather furnishes a new topic in the philosophy of logic, especially for either analysis and representation of knowledge systems (i.e. for rational agents). In par- ticular, the role that this formalization plays on the epistemic basis, and the related interpretation for rational agents will appear clearly. In such a frame the main concern will be to show how a knowledge frame, intended as the complete representation of an agent’s knowledge content, can be extended, and how it can be updated by means of a formalized notion of information. The explanation and formal definition of this epistemic concept is therefore the core of the entire research: it is obtained by understanding the inner conceptual difference between the notions of knowledge and information, describing the latter in terms of an essential relation between the user, its epistemic state, and the conditions for stating knowledge. This description offers moreover the basis for a constructive model of dynamic reasoning. The ability of a rational agent to make use of informational contents rather than referring to explicitly proved contents allows for submission to revi- sion: this can be seen as a procedure of type-checking, and it shows the essential connection to the decidability of forms of judgement. I maintain this result to be an important step towards the development of a system- atic treatment of errors in the constructive setting, and it introduces the possibility of multi-agent systems, merging and decision-making processes, another area which needs to be faced within the community of logicians inspired by constructivism.1 The structure of this chapter makes use of dif- ferent conceptual references; in particular, it is based on a possible world semantics and it includes the typical formalization of Kripke models for 1 See Primiero (2006).
- 20. Introduction 5 intuitionistic logic. The development of the procedure of extension and up- dating is obtained by making explicit reference to the distinction between analytic and synthetic judgements in CTT, which obviously has a quite important consequence in determining what can be accounted as a syn- thetic procedure. This is the connecting point to the problem of analyticity for logical processes. The idea explained and supported throughout the re- search is that the relation of logical consequence formalized within CTT, i.e. derivability in the constructive sense, provides ways to formally clarify its synthetic nature. This last topic is conceptually developed in Chapter 4 by showing how the principle of synthetic extension of logical reasoning regards essentially two aspects: (1) the structure of hypothetical reasoning and (2) the con- structive notion of meaning. The distinction between knowledge and infor- mation is not just a question of formal expressions, rather it is reflected in the conceptual frame: knowledge is based on the analytic development of the derivations—it is therefore characterized by the property of correctness and it provides the meaning of the concepts involved; on the other hand, the substrate of these procedures will be shown to be synthetic, represented by the concept of information: it is characterized as a procedure of conceptual change, in terms of the meaningfulness of the notions involved. The re- sulting theory of meaning is a coherent extension of the normally intended intuitionistic one: it does not contradict the insight of the meaning is use slogan, and it provides moreover a complete understanding of those cases which appear problematic to this view. This is obtained by an epistemic description of some formal elements and their operations: the notions of presupposition (and therefore a reformulation of its theory), assumption, and type declaration are the core of the theory here presented. This last chapter completes therefore the essential aim of the research, i.e. to match a new reading of the formal structure of CTT with a conceptual interpre- tation of the notion of knowledge and meaning. The hope is that what is presented here proves useful to a complete understanding of the constructive philosophy of logic and to a general view on knowledge processes. The possible developments of this epistemic description are various, and involve at least two important topics. The first concerns the nature of some particular kind of logical objects, which can be analysed directly in relation to their informational content: this applies in particular (at least in the constructive setting) to abstract entities as concepts, functions, and types. The sense in which the word “abstract” is here used is of a peculiar kind; it does not refer to abstraction as non-concrete, or non-definable, or un- able to produce effects. The notion of information here considered can help in understanding the nature of such entities. This is to my mind an open field of research for extending the philosophical basis of constructivism. The second topic concerns more directly the philosophy of information in con- nection to its ethical problems and open questions: the present definition of
- 21. 6 Introduction information (and in particular its “weak” epistemic status) reveals the ethi- cal consequences of the (here rejected) procedure of accepting informations as knowledge-contents to which truth-values can be ascribed. This intuition is particularly fruitful in describing phenomena of collective acquisition of “false information” via media and informational systems. At the end of this introduction, I want to express gratitude to the peo- ple who have been my guidelines, in these years of formation, study, and personal growth: Giuseppe Roccaro, who introduced me to the beauty of logical reasoning, especially by the words of the Greeks and Latins—my knowledge and my scientific development owe a lot to him; Göran Sund- holm, who during my year spent at Leiden Universiteit and since then until now made me see the other way than realism, gave me the comfortable feel- ing of studying something which exactly fits with my perspective on logic and other things—in the last years he has been the fruitful discussant of my ideas and the opponent of every word of mine, and has been for me a moral support and an incredible human help; I owe gratitude to Per Martin-Löf for having accepted to follow my studies during a semester spent at Stockholm University, posing crucial philosophical questions and illustrating the tech- nicalities I needed to know in order to systematize this work—it has been a personal and human pleasure to know him and to learn from him. I am personally responsible for every conclusion I have drawn from papers and notes which he did not yet decide to publish; professor Leonardo Samoná, my PhD coordinator, has done everything possible to let me pursue further my studies in the best conditions, showing a great trust in me; Giovanni Sambin, in some short meetings, gave me his personal insight into construc- tivism as a way of doing and thinking. I owe a lot to other people: my parents, my sister, the rest of my family, and Mirjam; everyone at Dep. FI.ERI at the University of Palermo and the people at the LabLogica Group; Giuseppe Rotolo; the guys and col- leagues at Biblioteca Tematica “Potere e Sapere”; the people at the Facul- teit Wijsbegeerte, Leiden/Amsterdam/Delft—Dr. Catarina Duthil Novaes, Dr. Maria van der Schaar, Dr. Arianna Betti, Dr. Bjørn Jespersen; the peo- ple at the Philosophy and Mathematics Departments in Stockholm. Many others should be mentioned here, because three years of life are long and full of experiences.
- 22. 1 Constructive Type Theory: Foundation and Formalization 1.1 Philosophical Foundation Constructive Type Theory has been developed by Per Martin-Löf in a series of papers and lectures since the 1970s: its first formulation, known as Intu- itionistic Type Theory, was based on a strong impredicative axiom which allowed a type of all types being at the same time a type and an object of that type; it was abandoned after it was shown to lead to contradic- tion by Jean Yves Girard; the reformulation of the entire framework led to a strong predicative theory, which is now known as Constructive Type Theory (CTT). The theory has its theoretical core in the contribution by Brouwer and Heyting to Intuitionistic logic, and it is therefore built on a constructive epistemic framework, providing a new interpretation to many of the central notions of classical logic, such as those of proposition, truth, and proof. I will begin by presenting in this section some general aspects of the constructive type-theoretical approach, analysing in the next sections its formal structure. To start with, only a general theoretical description of such a logical approach will be given and later fully explained, especially in connection with the notions of judgement and proof. The main aim of the present chapter is thus to present the theoretical, logical, and formal basis of CTT: a philosophical analysis of the theory and the explanation of the elements allowing to reconsider the problem of analyticity in the light of the constructive framework will in turn justify the introduction of the notion of information within the epistemic description. In the first instance, it should be stressed that the theoretical approach at the basis of CTT does not amount to a meta-mathematical interpre- tation: following Heyting’s work, the theory starts instead by giving the constructive reading of the notion of proposition. It does not begin with a formal axiomatization and a mathematically formalized semantics: rather, one explains what a proposition is, what it means for a proposition to be true, and when one is allowed to assert the truth of a proposition, in 7
- 23. 8 1. Constructive Type Theory: Foundation and Formalization order to verify what one can truthfully derive from it (i.e. which acts of inference preserve knowability of truth). Propositions are in turn explained in connection with the act of knowledge asserting them, namely, the act of judgement. In this first rather obvious sense, meaning is given within the type-theoretical framework in terms of computation, defining syntax to form canonical expressions, describing how assumptions-free judgements and hypothetical judgements (judgements made under assumptions) are formed: the meaning of each proposition will be given by the knowledge of a method to establish its truth. This systematization of the theory is based on the role given by Martin-Löf to logic and mathematics: logic is intended as the art of reasoning in a very old-fashioned sense, namely, the one intended by the Greeks and the Latins. Under this interpretation logic is complementary to mathematics, the latter being directed to prove the- orems, whereas the activity of a logician is to build formal languages by means of forms of judgement and inference rules to obtain those theorems searched by mathematicians. Once logic is not only based on a purely for- mal interpretation but is also used as a proper theory of reasoning and knowledge, it regains its status as the foundation of scientific knowledge, connected both to philosophy and mathematics: logic is not just an empty formal structure in the Hilbertian style, but is rather thought of as an interpreted system, whose objects are filled with meanings.1 This approach refers thus not just to a mathematical theory, but rather let us refer to it as a logical framework, in which different philosophical problems are investi- gated. At the same time, the framework is a useful and powerful technique for both mathematics (logic intended as proof-theory or meta-mathematics) and computer science (symbolism to design programming languages). It is an essential aim of this work to develop further the use of CTT as a the- oretical and logical framework, in order to consider and to solve a specific epistemic problem. 1.2 Basic Epistemic Notions Constructive Type Theory is to be presented first of all as a theory of expressions in the old sense, comparable to Aristotelian and Stoic logic. Aristotelian logic developed the forms of reasoning by means of judgements in the form “S is P”, S being a schematic letter for the subject and P for the predicate, analysing all the possibilities composed by affirmation, negation, universal and particular judgements, and using syllogisms as forms of infer- ence. This schema was completed by the Stoics, by introducing consequence as a form of judgement (“If A then B”), plus disjunction, conjunction, and negation. Aristotelian logic was pervasive and was in fact the only one until the 19th century; the work of Frege represents at the same time the first 1 Martin-Löf (1993) presents this idea of the essential connection between logic and mathematics.
- 24. 1.2. Basic Epistemic Notions 9 modern formalization for a logical calculus and the original ancestor for the notion of type2 ; on the other hand, Gentzen notoriously provided the first analysis made on the basis of sequents, using introduction and elimination rules. These essential notions of modern logic appear at some stage and with different roles in the formalization and methodology at the basis of CTT. I will start considering the epistemic notions used by the theory, developing them in connection with the proper logical structure and formalism. In later sections also the historical foundation will be presented. The essential innovation given by the constructive approach is the new interpretation of the conceptual connections between the notions of: • Proposition • Truth • Falsity • Knowledge.3 These are key notions for the philosophical setting of the theory; their understanding relies on the concept of judgement, which allows the con- nection of the notion of proposition with those of truth and falsity, with affirmation and refutation being the form of construction of a judgement,4 as follows: • A is a proposition. • A is true. • A is false. The notion of judgement is epistemically defined by saying what it is that one must know in order to have the right to make it: this means that from an epistemic perspective a judgement is a piece of knowledge. It is the aim of this research to explain what knowledge is, and which judgemental forms can be properly considered knowledge candidates in a constructive framework. This explanation is thus given according to the philosophical basis of Intuitionistic logic: at this stage the general notion of evidence can be used, as the one which (the Intuitionistic concept of) knowledge is based on. A sketch of the conceptual relation of these basic terms is the following: evidence → (correct) judgement → knowledge These are the basic epistemic notions, completed by their non-epistemic counterparts, namely, the notion of proposition, and the alethic notion ascribed to it, i.e. truth and falsity5 : 2 Sundholm (1986) underlines in which sense the logic at the basis of Martin-Löf’s Type Theory represents a return to the Fregean paradigm. 3 Cf. Martin-Löf (1995). 4 Martin-Löf (1995, p. 188). 5 Cf. Martin-Löf (1995). The distinction between proposition and judgement will play an essential role throughout the formalization of the theory, and for the understanding of the philosophical problem introduced in Chapter 2. A first brief
- 25. 10 1. Constructive Type Theory: Foundation and Formalization Epistemic notions Non-Epistemic notions evidence truth-maker judgement proposition correctness truth/falsity knowledge state of affairs where evidence is to be intended as the basis on which a judgement is knowable or a proposition established as true (its proof). In turn, to give a proof of a proposition allows to assert the judgement which says that the proposition is true. This implies of course that in order to state that a certain proposition A is true, one has to construct its proof (say a), so that “A is true” is equal to “there exists a proof a of A”. Of course the notion of existence which is used here to define the one of truth is something other than the notion of existence ruled by the existential quantifier6 : it is related to the description of what was explained by Aristotle as existence of an essence, or by Frege as existence of an object which falls under a concept. According to this interpretation, the existential quantifier depends on the more primitive notion of existence, like when one affirms that (∃x ∈ A)B(x) is true = Proof(∃x ∈ A)B(x) exists, a formula in which this distinction is obviously clarified by the presence both of the quantifier ∃ and of the verb “exists”. It is only at this point, in virtue of the constructive explanation of existence as “instantiation” that classical logic is rejected.7 Thus, the theory relies on the general Verification Principle of Truth, according to which truth is justified by the existence of a proof of the proposition, which makes the concept of truth for proposition no more primitive, but rather defined: Principle 1.1 (Verification Principle of Truth) The notion of truth is defined as existence of a proof (Truth = Proof + Existence). A summary of the crucial points of this (general) theoretical-foundational approach is the following8 : explanation of this connection can be given here as follows, for the sake of clarity: asking what a proposition A is means nothing but asking what one needs to know in order to assert the judgement “A is a proposition”. Here comes the Intuitionistic understanding of the notion of proposition, via the explanation of the meanings of the logical constants; given these explanations, a certain proposition A will be given by the set of its proofs. In this way, a proposition is defined by stipulating how its canonical proofs are formed. 6 For this explanation cf. Sundholm (1993, 1994). 7 It is relevant to underline the importance of the analysis developed by Martin- Löf (1991) relatively to the notion of logically possible existence and actual ex- istence, a topic that will be reconsidered later. A formulation of existence as instantiation is given by Martin-Löf (1992). 8 This list is extracted with some variations from Sundholm (1993).
- 26. 1.3. Types 11 1. Propositions are explained in terms of the proofs which are required for their truth. 2. Proofs are constructions. 3. Constructions are mathematical objects. 4. The theorem (justified judgement) “the proposition A is true”, in its explicit form, sounds: “the construction a is a proof of A”. 5. A theorem is explained by virtue of what is necessary to know in order to make that judgement. 6. Propositions have provability conditions (whereas judgements have assertability conditions). 7. Judgement and correctness for judgement are epistemic notions, propo- sition and truth/falsity for it are alethic notions. 8. Truth is given in terms of the existence of a proof. After this presentation of the main framework of the theory, the analysis of its conceptual and formal basis follows: this will be done by starting from the philosophical problems endorsed by the theory, developing the logical formal structure, and paying particular attention to the Intuitionistic framework of the theory. 1.3 Types 1.3.1 Constructive Notion of Type The notion of type in use within CTT has deep conceptual and formal roots in the history of logic.9 The constructive notion of type can be possibly explained in connection to different general terms, all of them well known in the development of philosophical and mathematical logic, such as: – Category – Type (classical version) – Sort – Level The notion of category obviously recalls first of all the use of this term in the Aristotelian logic (κατηγορία), and the form of predication conveyed by the judgemental form is essential to the understanding of the present frame- work, because it represents the essential root of the type-theoretical formu- lae. The corresponding Aristotelian notion represents the meaning-giving term in every well-formed predication: κατηγορία comes notoriously from the verb κατηγορείν, abbreviation for the long form κατά τινος ἀφορεύειν, “to say something about something”. Within Aristotelian logic and meta- physics, there is an essential relation between what a being is, namely, its 9 The background of the intuitionistic notion of type is presented by Martin-Löf (1987, 1993).
- 27. 12 1. Constructive Type Theory: Foundation and Formalization essence, and the predications being performed in relation to it: if essence corresponds to meaning, the latter is not just given by the category of sub- stance (οὐσία, the first of the categories); rather categories determine all the meaningful predications which can be performed in relation to the subject involved. Thus, the (correct) forms of predication built up by the copula scheme “S is P”10 are the ones which illustrate a thing’s essence,11 and categories are in this sense the way meaning is preserved. In relation to the mentioned connection between ontology and predication, Aristotle explains categories according to a twofold direction, as categories of “what is” and categories of “what is said”.12 The verb “to be” in its form “is” (copula) inside the Aristotelian form of judgement “S is P” is not a 2-place relation, but a way to attribute the category P to the subject S, and this suggests a rather obvious similarity to the notion of type as intended within CTT, in which the identity between propositional predication and set-theoretical properties fully and explicitly holds.13 In particular, forms of predication for this theory correspond to instantiations of a certain type with one element, which means exactly that a certain individual belongs to a certain class: thus, the predication in the type-theoretical formalization will be in general represented by a subject predicated within a certain type. The connection between the Aristotelian notion of category and the constructive types is quite evident, both being essentially meaning-giving structures.14 The notion of category as intended by Aristotle is radically changed by Kant. The use Kant made of this term in the Critique of Pure Reason is related to a pure concept of understanding, which in turn corresponds to a form of judgement. The distinction with the Aristotelian notion of category is evident: the linguistic category is not extracted by being recognized in what there is (ontology), but rather from what is thought. On the other hand, the correspondence to categories as meaning-giving forms of expres- sion is still entirely preserved under the Kantian view, and in turn it is even stricter with what later we will determine as proper categories of Type Theory.15 10 This one represents already a rough translation of the proper form conveyed by Greek language; in fact, ὑπάρχειν reads more exactly as “belonging” leading to a formulation of the judgemental form as “P belongs to S”. 11 Martin-Löf (1993, p. 38) refers to the connection between the Aristotelian τὰ σχήματα τῆς κατηγορίας and the syllogistic schemes of reasoning, a link which is expressed, for example in Metaphysics, book ∆. 12 Aristotle (Cat, par. 2). 13 This is the “propositions-as-sets” interpretation, to be introduced later. 14 It is important here to underline that despite the mentioned similarity the use of the term “category” will be reserved later for a different kind of expression than what is intended by “type”; such a distinction will become natural by considering the question of method and particularly evident by means of the formalization. Cf. in particular Sections 1.3.4 and 1.5.3. 15 Moreover, in connection to the Kantian philosophy of logic, CTT has a central point in explaining the difference between analytic and synthetic judgements,
- 28. 1.3. Types 13 The notion of category as a meaning-giving structure in the context of predication was explicitly restored by Husserl: expressions are considered by Husserl as meaningful signs, and meaning categories describe in turn as categories of the possible objects referred to by the expression, being also possible for a meaning category to be empty of real existing objects. Thus, the Husserlian system distinguishes clearly between semantical and ontological categories, by making the two levels already involved by the Aristotelian treatment more rigorous, where language is the way of refer- ring to entities. Husserl considers both types of categories as essences, to be grasped by acts of thoughts; the study of essences is done in terms of essential insights on meanings and independently of the corresponding ontological kinds. Essences are distinguished between formal essences (cate- gories), by means of which individuals are described, and material essences (regions), classifying entities according to their nature.16 In the analysis of categories and types, the original link between the linguistic and the ontic regions will be restored, and this will directly determine the nature of the method and of the syntactic/semantic distinction for CTT. A different use of the notion of type was notoriously due to Peirce,17 who introduced the distinction between token and type. The latter term refers to the shape or form of something, whereas the former means the differ- ent occurrences of such a form. Referring to this terminology, the notion of type introduced by Russell18 was somehow unlucky, referring to the word “type” in a different way: in fact, such an understanding of the term type has its own roots in the notion of function, essentially based on the Fregan understanding of this concept. CTT thus represents the evolution of the notion used by Frege, and our notion of type represents a strucure playing the role of categories and corresponding to formal rules holding for func- tions. In Section 1.6 the structure of the theory will introduce the notion of dependent object, and to this aim it is necessary to explain the technical connection of types with the structure of functions: a brief historical and technical introduction to the development of the notion of function will be given there. Meanwhile, it is here relevant for the clarification of types to give some insights on its intuitive notion: one generally refers to a function as a procedure that provides a value for each element given to it as input. The relation can be either a mathematical formula or a syntactic method, deter- ministic in that it has to produce always the same value on the basis of the same argument. Frege, on the Aristotelian assumption that the main cate- gory for each object is τό τι ἕν εἶναι, the substance, started by trying to use a unique universe, the one of objects (Gegenstände), and developed his sys- tem by making use of functions, to be able to go from objects to objects, and something that will result later extremely important in our analysis. Cf. Martin- Löf (1994). 16 Husserl (1913a,b). 17 Peirce (1906). 18 Russell (1908).
- 29. 14 1. Constructive Type Theory: Foundation and Formalization expanding the system by applying functions to functions: in the Grundge- setze der Arithmetik, Frege established that the formula (α1, . . . , αn) is the “type” (using the modern terminology) of n-place functions, which gives rise to an object of a specific level (Stufe).19 This means that if α1 up to αn are types, we can form a new type which collects all the previous ones (here we already introduce a formalization for such a predication): α1 : type, . . . , αn : type (α1, . . . , αn) : type A schema of the correspondence with functions is the following: () object (Gegenstand) (()) unary function (()()) binary function (() . . . ()n) n-ary functions which take many functions into an object (Wertverlauf ) where clearly unary functions have objects as arguments, secondary func- tions have unary functions as arguments, and so on. Later, in his Begriff- schrift, Frege introduced the judgeable contents (beurteilbare Inhalte), con- sidering propositions inside the universe of objects: this gave rise to anti- nomies due to impredicativity. Russell (1903) presents a way out from the paradoxes generated by Frege’s functional hierarchy, and in this sense it represents the natural ancestor of Type Theory. The Russellian type theory is related to the Fregean functional hierarchy by accounting the “simple” types, independently from the complexity of definition (so that it naturally reflects the order of “objects”, “concepts”, “second order concepts”, etc.).20 Russell presented the simple theory of types in two appendices (1903), then developed the ramified version (1908): in this new version the type of a function depends not only on the types of its arguments, but also on the types of entities referred to, and quantified over, by the function itself, i.e. through typing propositions. In the simple theory of types Russell mod- ified the Fregean structure by defining: – The type of the individual valued functionals – The type of proposition-valued functions Referring again to the previous case, when α1, . . . , αn are types, the Russellian theory understands [α1, . . . , αn] as a type too, i.e. n-ary propo- sitional functions with types α1, . . . , αn as arguments represent types themselves: α1 : type, . . . , αn : type [α1, . . . , αn] : type 19 The level of a type is defined by Frege (1884) as the maximum of the levels of the argument types plus one. 20 One should also remember the anticipation of the simple theory of type due to Schröder. Cf. Church (1976).
- 30. 1.3. Types 15 and the system is then enlarged adding clauses for relational types: [] = type e.g ⊥, [[]] = type(type) unary function, e.g ¬ [[], []] = type((type), (type)) binary function, e.g. ∧, ∨, ⊃ [[[]]] = type(type(type)) quantified function, e.g. ∀, ∃ [[] . . . []n] = type(type) . . . (type)n type of n-ary functions/relations Both structures, the Fregean and the Russellian, are powerful enough to express systems of a certain complexity, such as in the context of first-order logic, but not enough for more complex systems. A new notation for the theory of types was then introduced by Schönfinkel in 1924, based on the idea of representing functions of n arguments as a unary function having a value corresponding to a function with n − 1 arguments, proceeding until one reaches the ground types (individuals and propositions). In this way, it is possible to give three clauses for forming types: 1. ι (for individuals) is a type. 2. o (for propositions) is a type. 3. If α and β are types, then (βα) is a type (with α for the argument type and β for the value type). Accordingly the level is defined in the following way: L(ι) = L(o) = 0 L((αβ) = max(L(α) + 1, L(β)), which represents the basic structure for Church’s notation, and for the structures developed by Schütte, Curry, and Ajdukiewicz.21 In general, the simple type structure makes it possible to type all the constants of first- order logic, while with dependent types of Intuitionistic type theory one is able to type even quantifiers whose domains vary. According to Martin-Löf something is never an entity without being of a certain sort or kind, and each mathematical object is always typed: such types (as we will see later) are the source of the categories of predication, giving rise to the syntax and semantics of the theory. Whenever the notion of type is understood in this deep and broad philosophical aspect, being assimilated in a general and intuitive sense to a structure constituting a family of objects determined by any property, together with an equivalence relation, the resulting formal theory is of a specific kind: such a notion of type is conceptually prior to, and provides an interpretation for, other notions such as the one of proposition, or the mathematical ones of set, elements of a set, the set-valued functions over a given set, and predicates over a given set. Thus, a theory of types can be used to present a theory of sets, using variables ranging over sets and higher-order objects, but in fact by choosing to use the more general and basic interpretation of the notion 21 Martin-Löf (1993) has treated the modern evolution of the notion of type.
- 31. 16 1. Constructive Type Theory: Foundation and Formalization of type, one understands the theory as a general logical framework able to formalize expressions, as it has been done at the beginning of this section: this kind of type theory is usually referred to as the monomorphic version of the theory, whereas starting by defining the types of sets (or proposi- tions), the set-formation operations, and the proof rules for these sets, one considers a specific type and thus refers to the polymorphic version. In the monomorphic version the notion of set can therefore be intended in all of its generality, allowing to consider a logical procedure such as assumptions on sets not yet defined.22 The monomorphic version of the theory allows for the introduction of different notions (sets, propositions, and similar) in terms of types; moreover, it leads to formalize derivations by means of metavariables ranging over formulae, and it requires the explicit formulation of all the information on which arguments are based: an application func- tion on two sets will, for example, take two arguments in the polymorphic version (i.e. a function from A to B and an element in A), whereas the fully explicit formulation of the monomorphic version will take four arguments (respectively, the two sets A and B, the function from A to B, and finally the element in A).23 Starting with his early work (1975), Martin-Löf has developed his type theory in a purely predicative way, so that second-order logic and simple type theory were not to be interpreted in it; the theory presented in his later publications (1982, 1984) is polymorphic and exten- sional, and the semantics given for the normalization procedure which lets an element be computated to its normal form provides a strong elimination rule, needed for propositional equality, in a way that judgemental equality is no longer decidable. In order to overcome this problem the monomorphic version is used, in which the equivalence relation needed by the definition of type and given in order to state the identity between objects within a certain type is decidable. Therefore, great attention has to be given to the notion of identity involved and to the formal rules for it. In Section 1.3.2 I proceed in defining the monomorphic notion of type, by considering the general expressions that will provide the basic relations between types and their objects. 1.3.2 Definition The epistemic basis of CTT develops the notion of type in terms of its definition, by clarifying the relation between objects-of-types and types themselves. As it is well known, Aristotle underlined the strong connec- tion between definition (ὀρισμός) and essence (τό τι ἕν εἶναι), the former being the expression which signifies the latter, its λόγος.24 This amounts to a distinction in the clarification of the notion of definition itself: 22 See, e.g. Nordström, Petersson and Smith (1990), pp. 137–138. 23 In Section 1.9 we will insist more on the role of informational content for the distinction between the monomorphic and polymorphic versions. 24 See, e.g. Arsitotle (Top, 101b39).
- 32. 1.3. Types 17 1. Real definition is intended as a genuine explanation of meaning. 2. Nominal definition is intended as an equational or identity definition. To give a real definition means to express an analytic recollection of all the (definitional) properties of a term, whereas to give a nominal definition means to establish an equational definition between such a term and some other sign. Defined expressions receive meaning by a nominal definition, while primary expressions derive meaning from a real definition. This dis- tinction is completely reflected within the type-theoretical framework: the definition of a type is given in terms of a meaning expression, being types of primary objects of the theory defined through the primary forms of judge- ment (the same is true for notions like object or family of types); on the other hand, definitions of other elements like class, relation, connective, quanti- fier are given in terms of defining equations.25 For this reason, within the constructive type-theoretical framework a real definition is a concept expli- cation, and can be understood as a conceptual analysis.26 The notion of type, obviously the first to be defined, is abstracted by the initial step of the theory, namely, by exposing a general theory of ex- pressions. There are four forms of expressions introduced by the theory, asserting respectively that27 : 1. A certain object is a type. 2. An expression is an element of that type. 3. Two expressions are the same inside the same type. 4. Two types are the same. The semantics of type theory explains what these judgements mean. In this way, to introduce and define a type one must know: 1. What it means for an object to be of a certain type 2. What it means for two objects to be the same within a certain type and they represent respectively what is called application criterion and identity criterion, according to the terminology introduced by Dummett (1973). The order in which these assertions are stated reflects the logical structure according to which the existence of a type comes conceptually be- fore the assertion that something belongs to that type; nevertheless, clearly the definition of any type is given according to some object belonging to it. In this sense the form of expression . . . is of the type . . . has to be preceded by (presupposes) the assertion that . . . is a type, 25 Martin-Löf (1993, pp. 60–61). 26 Cf. Sommaruga (2000, p. 2). The formal treatment of the notion of identity, given in Section 1.4.1, will say more on this essential topic. 27 Cf. Section 1.5 for further explanations and formalization.
- 33. 18 1. Constructive Type Theory: Foundation and Formalization where, for example, some α will take the place of the dots. This remark is necessary in order to introduce two problems: 1. Conceptual priority 2. Impredicativity The first reflects the theoretical structure underlying the theory, which will be explained in the following paragraph; the second is the well-known prob- lem caused by the Fregean hierarchical structure, avoided in Type Theory via the conceptual priority of types over objects belonging to types, and the essential introduction of the notion of category. 1.3.3 Conceptual Priority The foundation and systematization of the theory is done by setting an order for the basic notions introduced, determining a conceptual priority among them.28 Such a structure can be thought of as developing the Aristotelian πρότερον and ὕστερον κατὰ τὸν λόγον for the theory, the methodological and ontological distinction later translated by the scholastic tradition as prior and posterior secundum rationem. Involved in such a relation of order are of course the elements occurring in predications and the distinction between concepts defined or taken as primitive in the theory: this conceptual order determines a definitional order, established according to the nature of the objects to be defined; and finally, because a definition is an explanation of the essence (real definition), an order will hold also between essences. The following schema shows the sequence of priority between orders: Conceptual order ↓ Definitional order ↓ Essential order In the history of philosophy, in line with the mentioned Aristotelian dis- tinction and its scholastic explanation, Augustine’s De Ordine represents the medieval development of the Platonic inspired distinction between ordo intellectum and ordo rerum, whereas the Aristotelian tradition is followed by Thomas Aquinas.29 These are the ancestors of the priority between orders holding in CTT, which takes into account the order of things and their definitions as distinguished from the order of concepts. The conceptual order within CTT thus establishes the priority between the basic logical concepts of 28 Martin-Löf (1984, 1987, 1991, 1993). 29 Martin-Löf (1993, pp. 61–65).
- 34. 1.3. Types 19 • Proposition • Truth and the mathematical ones of • Set • Element of a set • Function The first two notions are connected by the concept of existence, namely, via that of proof. Existence of truth in terms of evidence is moreover developed by introducing the classical distinction between the categories of actuality and potentiality; thus in turn truth is explained as actual truth and potential truth: “Actual truth is knowledge dependent and tensed, whereas potential truth is knowledge independent and tenseless”.30 The actual truth of the proposition A, according to the Intuitionistic frame- work, presupposes a construction already obtained for A, while potential truth is the possibility to construct such a proof. Following the Aristotelian metaphysics, actuality precedes potentiality in the order of the real (i.e. in the order of entities).31 The notion of actuality corresponds of course to the instantiation of an act performing and realizing truth: here one finds the first theoretical justification for defining the logical notions of proposition and truth upon a more fundamental one, precisely the notion of judgement, which immediately states the distinction between the act of judging and what is judged. On the other hand, the mathematical concepts of set and element of a set are essential in that they represent an exact mathematical interpretation of the corresponding notions of type and element belonging to a type; the system is extended via the concept of function, which is the mathematical way to explain the relation between two elements belonging to equals (or different) types: on the basis of the Curry–Howard isomorphism (to be explained later in Section 1.5.7), the same is true respectively for the notions of proposition and proof. But in the first instance the order of conceptual priority holds between concepts and their definitions, i.e. the order can be established between two concepts32 : 1. If the understanding of a concept presupposes the understanding of the other concept 2. If the definition of a concept refers back to the definition of the other concept 30 Martin-Löf (1991, p. 143). About potentiality as possibility, in connection to the framework of CTT; cf. also Löhrer (2003). 31 Martin-Löf (1990), mentions the Thomist formulation “Actus est prior poten- tia”. 32 Sommaruga (2000, p. 5).
- 35. 20 1. Constructive Type Theory: Foundation and Formalization In this sense, it is clear that one establishes an order between the concepts treated up to now, in the following way: proof judgement =⇒ + proposition =⇒ truth existence The notion of judgement comes first because it should be understood as a “ground notion”, explaining on its own the concept of proposition as its content; the concept of proof is considered as the (proof)object instantiating a demonstration act for the proposition contained in the judgement. The assertion performed in a judgement regards the truth of a proposition. Thus, the first schema has a second extension, that does not rely anymore on the specific content of a proposition with its proof object: judging act (demonstration) → evidence → correctness ↓ propositional content The act of judging establishes a demonstration (proof not intended as ob- ject) which furnishes evidence for a propositional content, and gives rise to correctness for proofs. The problem of definition and the structure of con- ceptual priority are thus essential to the theoretical frame of CTT, such that the theory represents an attempt to build each form of judgement starting only from the explanation of what a type is, and what it means for an object to be of a certain type. This represents the way in which types are defined and in which categories are introduced: the connection between forms of expression within the theory and the objects these expressions refer to is settled by the syntactic–semantic method. 1.3.4 Method The starting point to explain the formal and theoretical structure of CTT is to give the definition of what a type is, namely, by answering the basic question “what is a type?” in terms of the other one “what does it mean to belong to such a type?”. To answer these questions Martin-Löf develops a method which is called syntactic–semantic, consisting of two parts: (a) Syntactic: the sense of a primary entity (in that it belongs to a certain type) is given by the process of composition of the formal expression which denotes such an entity. (b) Semantic: the sense of that entity can be understood contextualizing the rules of composition applied to obtain the expression in the first part (a).
- 36. 1.3. Types 21 This method allows us to clarify the nature of mathematical objects by paying attention to the expressions denoting objects,33 because these show exactly their meaning. Here one finds the connection between the notions of definition, conceptual priority, and identity, explained below: the rela- tion between an expression and the object it signifies represents the act of meaning or understanding. For an object to come into being the expression by which that object is denoted is necessary: the formulation of such an expression, consisting in the predication of the object within a type, repre- sents therefore the act of understanding the object. The connection between an object and its expression is thus a turning point for the method at the basis of the theory: a mathematical object is always expressed via the ex- planation of what is the type to which it belongs, and this brings us again to the conclusion that types come conceptually before objects, because the latter have an ontological status only if semantically typed, i.e. if their type has been previously declared. Also through the description of the syntactic– semantic method, the need clearly arises to justify the conceptual relation between the predication aptness of the type and its definition. The relation between the semantics and the syntax of the theory can be represented by a General Principle of Meaning, formulated in the following terms: Principle 1.2 (General Principle of Meaning) The relation between objects and expressions is given as follows: a certain object a is the meaning of the expression “a”; in the other direction, “a” is the expression denoting the object a. This principle reflects the natural direction from the ontological to the lin- guistic level.34 Thus, the syntactic level goes from the object a to its ex- pression “a”, and this means to consider the object in a purely formal way, the formalization consisting in divesting the object of sense, in the Hilbertian style. On the other hand, the process of endowing the expression “a” with sense means to give its content, referring to the object a. As the General Principle states, “a” is the expression of a, and a is the meaning of “a”, where an expression is obtained by the process of formalization. In such a process types are turned into type expressions, and objects into object expressions, so that the object set is turned into the category of set expressions (which is in turn its syntactic category). It is quite clear that the syntactic–semantic method is more than a simple distinction be- tween syntax and semantics: the ontological basis on which the theory is 33 Martin-Löf (1987). 34 Here Martin-Löf refers to the Husserlian approach, according to which “in naturliche Einstellung wir sind gegenständlich gerichtet”. Husserl considers the difference between the object a and its expression “a” by using respectively the expressions Bedeutung and Ausdruck. Moreover, in illustrating these notions dur- ing his lectures (1987), Martin-Löf refers to the Husserlian expressions Syntak- tische Kategorie and Bedeutungskategorie, while in (1993) he uses the Husserlian terms Sinnbeseelung and Sinnentleerung.
- 37. 22 1. Constructive Type Theory: Foundation and Formalization built and the development of its linguistic level allow us to understand the entire method as nothing but a sort of duality recalling the philosophical distinction between form and content. In fact, the relation between “expres- sion” and “content” and that between “object” and “type” can be thought of as a modern mathematical version of that ancient duality. Plato first introduced the distinction between εἶδος and ὄν, which was only an aspect of the all-invasive primary dichotomy between τὸ αἰσθητόν (the sensible) and τὸ νοητόν (the thought, or what belongs to it); for Aristotle the way from the ὕλη (matter) to the οὐσία (substance) is given inside the όν (be- ing) through the essence, τό τι ἕν εἶναι, namely, referring to “things in that they are things” (τὸ ὄν ἤ ὄν), an expression which in turn explains what ontology is about, and which we will take into account later. The mentioned dichotomy was then restored by the Scholastics in the terminology materia and forma, their connection giving rise to the substantia. Here the role of definition is particularly important, determining what really is the τὸ τί ἐστι (quidditas) — the being which really exists — the connection of form and content. Within the type-theoretical frame the relation between the construction and the object is given through the connection of form, repre- sented by the type35 : here we find the essential concept that mathematical objects are objects of knowledge which need to be expressed in order to be grasped. The syntactic–semantic method used by Martin-Löf is thus built up by the relation between the expression intended as object of a syntactic category and its meaning, i.e. the object for which the expression stands for, intended as a semantic category. The syntactic–semantic method is a way to state a new theory of essences, building a bridge between the seman- tics and a proper ontology. This method is enough for building up a theory of mathematical essences, given that in this interpretation a mathematical world of objects can exist only if expressed. The question which now natu- rally arises is the following: how many kinds of categories do we have? The answer is obtained by reducing the (classic) schema Syntactic category ↓ Semantic category ↓ Objectual category composed of three categories, to the following one Syntactic category ↔ Semantic category where the objectual category conflates into the semantic one because the ob- ject represents the meaning of the expression “a”, i.e. a itself, and therefore 35 Martin-Löf (1993, pp. 163–168). Another reference is made by Martin-Löf (1993), to the Heideggerian couple of terms Zuhandenheit, which explains the use of tools without paying attention to their formal structure, and Vorhanden- heit, which instead refers to the use of tools on the basis of the knowledge of their form.
- 38. 1.3. 23 the semantics is actually the ontology the theory speaks about. This last point has a further explanation: ontology is intended not just as the science about the things of any world, such that these are objects of other sciences, e.g. physics. Ontology is all about “things in that they are things” (in terms of the Aristotelian definition): this means to take into account objects as they are defined, i.e. objects in terms of the concepts they express, or they are defined by. Thus, a proper object of ontology is a defined object, an object expressed with all its (essential) properties. Ontology in this sense, conceptually near to the Aristotelian way of understanding it, amounts to a study of objects with the concepts they contain, that in the type-theoretical setting means to express objects in terms of the types they belong to. Hence, we are again considering the only way objects can be taken into account, by referring to the expressions they are (correctly) predicated in: by means of language the syntax and the semantics of the theory are connected, and the study of the formal expressions of the theory introduces the categories of the theory (Section 1.5.3), already mentioned in connection to the question of meaning and the problem of impredicativity. The introduced distinction between the syntactic and the semantic level of the method explains a basic distinction inside the notion of meaning,36 namely: – Sameness of meaning – Identity of meaning This distinction is of course of the greatest importance for the notion of synonymity and requires an explanation of the concept of identity, to be analysed in Section 1.4. What is relevant to underline at this point is that, on the basis of the conceptual priority, identity of meaning and even same- ness of expressions (e.g. nominal definition) ultimately refer to identity of objects, as primary elements of the theory. The link between the syntactic and the semantic levels for the type-theoretical framework can be thought of as a two-way relation between objectual (or semantic) and syntactic categories: objectual categories a ↓ ↑ formalization contentualization ↓ ↑ “a” syntactic categories This schema is to be considered as a modified version of the one already proposed by Aristotle in the first chapter of De Interpretatione, where he explains the connection between the object, the related movement in the soul, and the expression for it, as follows: 36 Martin-Löf (1987). Types
- 39. 24 1. Constructive Type Theory: Foundation and Formalization παθήματα τῆς ψυχῆς σύμβολα → πράγματα and which will be used as a basis by the Stoics37 and the Scholastics.38 This idea will be later endorsed by the well known “semiotic triangle” which states the relation between object, expression and meaning: meaning expression → object According to Martin-Löf, the new schema with only two elements (seman- tic/syntactic categories) includes the Saussurian relationship between signifié and signifiant inside the signe, and in relation with the chose: signifié signifiant → chose while in the case of the Fregean relation between Bedeutung, Sinn, and Ausdruck, the schema reverses its arrows, in the following way: Sinn expresses refers to Ausdruck . . . Bedeutung The triangle schema39 shows the relation between the three essential stages, the mental, the verbal, and the real. The theoretical problem one needs to solve within CTT concerns either the necessity of establishing the third realm of concepts (e.g. as done by Frege) or the possibility of conflating together concepts (meaning, if linguistically intended) and objects, so as to make no categorical difference among them. The solution is given in a proper way by the notion of ontology explained above, which we present here as the General Principle of Ontology: Principle 1.3 (General Principle of Ontology) Categories of objects are actually categories of meaning, because essences of objects, i.e. things in that they are things, are expressed by concepts via their meaning. 37 They will change the words, using respectively τυγχάνω, σημαινόμενον, and σημαίνον. 38 They will translate the schema with the following Latin terms: res, pas- sio/intentio/conceptus animae or intellectus, and finally nomen. Martin-Löf gives references to Ockham, Boethius, Thomas (1993, p. 175–176). 39 This schema was originally presented in Ogden and Richards (1923).
- 40. 1.4. Identity 25 It is therefore essential at this point to introduce the topic of identity, both because it arose already in the conceptual framework of the theory and because it will be essential in introducing the formalization and the sort of type theory considered all along the rest of this chapter. 1.4 Identity The definition of a type is hence given by explaining what it means for an element to belong to a type (application criterion) and for two such elements to be identical within a type (identity criterion). The notion of identity is thus clearly involved at the core of the theory, both for the definition of type and for the theory of expressions. Moreover, identity was also implied by the notion of synonymity by introducing sameness of meaning or identity of meaning for expressions, and this will be again a central topic in Chapter 2, where the problem of analyticity will be presented. It is therefore essential to present the theoretical treatment that CTT gives of the notion of identity. The relation of identity between two expressions holds primarily when such a relation holds between their meanings (i.e. objects); therefore, iden- tity of objects (i.e. identity related to the ontological level) comes before the identity related to the linguistic level (synonymity). The way these notions are introduced in the framework is in connection with the schema of relations between syntactic and semantic levels of the theory, presented by extending the schema of categories to a four-element schema, which introduces the identical elements. If the original schema presents a two-way relation between the object a and its expression “a”, i.e. the relation be- tween the propositional (or numerical) expression and the type expression via the object itself, the synonymity of two expressions referring to the same object, and the identity between objects themselves, can now be in- troduced. For example, one can take the two objects a and b, equal to each other inside the type α, and construct a schema including their expressions, “a” and “b” (Figure 1.1).40 In this schema the relation of evaluation corresponds essentially to com- putability; for example, it holds between the object S(S(0)) (Peano’s clas- sical axiomatic translation of the Arabic numeral) and 2 (or the relative expression). The process of evaluation is particularly important here, be- cause it makes it possible to extend the previous schema, which included only two levels (syntactic-semantic), by introducing a third level represented 40 The following schema is built up from different elements stated and explained by Martin-Löf (1993, pp. 187–192). The example for this schema presented by Martin-Löf refers to the mathematical object 2 + 2 as the meaning (a), “2 + 2” as its expression (“a”), and the object 4 as the semantic value of 2 + 2 (b) and “4” as the proper expression for it (“b”). This schema includes the semiotic triangle as its proper part: in that case the evaluation is made only referring to the object level (semantic level), not to the syntactic level.
- 41. 26 1. Constructive Type Theory: Foundation and Formalization a b a b formalization signification Semantic Level Syntactic Level Meaning Expression evaluation Semantic value Syntactic value (identity between objects) (identity between expressions) Content Form Defined/non canonical Primitive/canonical Figure 1.1. Schema including syntactical, semantical and identity relationships inside the type-theoretical framework. by the identity relation: this is done in terms of definition. The relation of evaluation holds between the definiens and its definiendum, so that a is a defined term (always graspable through its type-expression “a”), which is evaluated as identical to b, the latter representing a primitive term. The definitional chain between a and b can be fulfilled by one step of computa- tion (if a is a primitive term) or else by more steps. Moreover, a being the sense of “a”, and representing b in this schema, the result of the evaluation process, namely the reference, one should be able to understand if sense and reference can be considered equal.41 Thus, the relation represented by the horizontal arrow in the upper half of the schema is the relation of evaluation or computation, which states the identity of meaning, or synonymity. It is possible to state the equality between sense and reference only at the level of objects, while they are syntactically the same (i.e. concerning identity of expressions, in the lower half of the schema) only if both objects are primitive ones, i.e. the definitional chain has zero steps. 41 This theme is developed by Martin-Löf (1993) in obvious connection with the Sinn/Bedeutung problem in Frege’s writings (cf. Martin-Löf 2001): in this sense the Fregean idea that the Sinn corresponds to the object including its expression or mode of presentation (Art des Gegebensein) is particularly relevant; the object is identified via such expression, so that identity of senses results in identity of expressions (cf. also the Husserlian idea in Ideen that the logical meaning is given via a certain expression (Ausdruck)); for a detailed explanation of the Sense/Reference distinction in CTT, see Primiero (2004).
- 42. 1.4. Identity 27 It is possible now to explain the identity criterion by distinguishing three versions of the concept of identity inside the theory42 : 1. Semantical identity criterion, which corresponds to definitional identity, or identity between objects (=) 2. Syntactical identity criterion, which corresponds to syntactically induced identity, or identity between expressions (≡) 3. Abstract or transcendental concept of identity Definitional identity (1) is introduced within the theory by the following rules: – Reflexivity – Symmetry – Transitivity – Substitution of identicals by identicals. The rules for reflexivity, symmetry, and transitivity are the common rules holding in mathematics, and their formalization for objects belonging to types and for types themselves will be shown in Section 1.5.2; the fourth rule, the substitution of identicals by identicals, allows to state the equiva- lence between definiens and definiendum within a definition. These remarks complete the introduction of the notion of identity, in addition to what has already been explained in relation to identity for expressions. The role of identity for expressions is also relevant in connection to the informativeness of the evaluation procedure, and this is important for the task of a critical analysis of the synthetic nature of the logical system.43 1.4.1 Definitional Identity vs. Syntactically Induced Identity The problem of definition is at the foundation of the type-theoretical frame- work, and certainly it presents a deep connection to the notion of identity. It has been explained how the notion of definition has to be understood, distinguishing between real definition and nominal definition. In the case of definitional identity, one is considering the level of nominal definitions: let us remember here that a definitional identity corresponds to a nominal defini- tion, which is the way to obtain the meaning of defined expressions. A nomi- nal definition represents then a stipulation for which no further justification is required, and it can be represented by the ancient couple “definiendum = definiens”.44 In the first instance, one distinguishes between definitional identity, which is a relation between linguistic expressions, and the relation of identity between the entities which are denoted by those expressions, 42 Cf. also Sundholm (1999). 43 Martin-Löf (1993, p. 236). 44 Ibid. (1987).
- 43. 28 1. Constructive Type Theory: Foundation and Formalization i.e. their meanings.45 The first kind of identity is given by the sign “=def ”, so that when one refers to a relationship such as a =def b it concerns the identity between two expressions. The second type of relation holds between the objects one is talking about, and it is formalized as a = b. The informal reasoning behind the interderivability of these two kinds of formulae is the following: given the identity axiom a = a, the formula a =def b implies that (a = a) =def (a = b), so that a = a and a = b have the same meaning and this immediately gives us the conclusion a = b.46 On the other hand, the identity of the objects a and b is enough to state the identity of the respective expressions, so that it holds the formula a =def b. Moreover, on the basis of the Intuitionistic approach, the validity of the judgement a = b must of course correspond to the possession of its derivation. If such a closed derivation is supposed to hold, then it is clear that the identity a = b implies the interconvertibility (formal counterpart of the informal “definitional equality”) of the terms a and b. In this sense we can say that two derivations are interconvertible if, and only if, the proofs that they represent are identical, so that “identical” means in this context “provably identical”. The relation of definitional identity is expressed by Martin-Löf according to three principles,47 each of them having a formal counterpart, namely, a conversion rule: 1. Definitional equality between the definiens and its definiendum: redex conv convertum(...) 2. Substitution of definitionally equal expressions for a variable in a given expression leads to definitionally equal expressions (preservation of defi- nitional equality under substitution): a conv c b[a] conv b[c] 3. Definitional equality is reflexive, symmetric, transitive: a conv a a conv b b conv a a conv b b conv c a conv c These are the general formal rules holding for the logical notion of identity and instantiated in the forms of judgements. Such rules allow the formula- tion of judgements either for a set-theoretical or for a propositional system, 45 Ibid. (1975a, pp. 101–104). 46 Ibid. (1975a, p. 102). 47 Ibid. (1975a, p. 93).
- 44. 1.4. Identity 29 via the expression of definitional identity between elements of a set and proof objects for propositions. In this way one understands the difference between definitional or semantical identity (a = b) and syntactically induced identity (a =def b): the first one is much stronger than the second one, in- volving the ontological aspect of the theory; syntactical identity refers only to the identity of the formal expressions we use to express objects.48 1.4.2 Identity as Theoretical Notion The identity issue in its philosophical and logical aspect has been widely considered since antiquity. It is clearly already present in Platonic dialogues, entering in the explanation of the relation of things to ideas; Aristotle brings this problem to its core, by considering the nature of essence, for which the notion of “being” is treated in connection to “predication”, thus referring to the categories of “sameness”, “otherness”, and “contrariety”.49 Identity is explained in terms of predication when Aristotle says that two things are identical if all that is predicated (or predicable) of one of them, is predicable of the other.50 The principle of identity (“a being is what it is”) is thus obviously central to the Aristotelian philosophy, and it notoriously expresses the positive formulation of the basic principle of contradiction: “a being cannot both be and not be at the same time and under the same respect”,51 the logical principle par excellence, both a principle of knowledge and reality. Since these first formulations, the identity issue and the relation to definition was a central topic in Scholastics, particularly in Ockham, with the distinction between definitio quid rei and definitio quid nominis.52 But one has to wait until Frege to find a fruitful connection for the development of the notion of identity with that of type: by using a unique universal type (the sort of all objects), Frege explained the question of identity both in terms of equality of content (Inhaltgleichheit) and as the binary relation of identity, respectively in the Begriffschrift and in the Grundgesetze; in Über Sinn und Bedeutung he explains the identity “=” as a relation between 48 Note that within the Hilbertian formalistic perspective, the semantical identity criterion can be only reduced to this syntactic criterion, because all expressions are empty of their meaning. In Sections 1.5.4 and 1.5.5 the ground types for sets (set) and propositions (prop) will be introduced; the notion of identity presents the following meanings within those types: – With A = B : prop the concept of material equivalence between two proposi- tions is intended. – With A = B : set the concept of equipotency between two sets is intended. 49 Martin-Löf refers in particular to Metaphysics, book ∆, 1018a35-39; cf. Martin- Löf, (1993, pp. 41–42). 50 Aristotle [Top] book 7, cap. 1, 152a 5–30. 51 Cf. Aristotle [Metaph], book Γ, 1006a 3–5: “ἡμεῖς δὲ νῦν εἰλήφαμεν ὡς ἀδυνάτου ὄντος ἃμα εἶναι καὶ μὴ εἶναι.” 52 Ockham (1324), Logica terminorum, III, 26.
- 45. 30 1. Constructive Type Theory: Foundation and Formalization signs, in order to introduce identity of reference for distinct signs, so to justify the concept of Sinn. The notion of identity then occurs in a pivotal role in the famous Quinean slogan “no entity without identity”,53 which can be explained in terms of the following statements: 1. No entity without type 2. No type without semantical identity The Quinean approach to identity is in order to avoid ambiguity: his in- terpretation of the sign “=” explains it as the extension from “is” (copula) to “is identical to”, and the identity conditions are stated as the “divi- sion of reference”. Thus, the problem of identity is stated in connection with synonymity for sentences (identity of meaning), analyticity, and infor- mativeness of identity sentences.54 According to Quine, identity assertions which can be considered true and useful are built up by different singular terms referring to the same thing,55 and this explains the difference between forms of predication which are expressible as a function “Fa” and those for which the sign “=” is required. In particular, Quine holds that synonymity between sentences can be explained via the notion of analytic sentence.56 In CTT the Quinean task to make these relations clear is actually obtained; the structure of reference and meaning is stated as follows57 : – An object expression stands for its meaning (which is the object itself). – A type expression signifies its meaning, which is a type. – Dependent or function objects or types have no value or reference. – The reference of an expression whose meaning is a non dependent or func- tional object results from the evaluation of the meaning of the respective expression. The identity of meaning of two expressions is given in terms of semantical identity: identity of meanings amounts of course to identity of the objects which represent the evaluation of those expressions. In general, in the case of CTT there is no type without semantical or intensional identity, and moreover also no such identity without type; then semantical identity is al- ready typed.58 The notions of identity and synonymity will first be formally considered in Section 1.5 and then taken into account again in Chapter 2: they represent the essential notions in order to clarify the idea of analyticity in logic and to identify the role of information for logical processes. 53 Quine (1958, 1960). 54 Ibid. (1960, cap. 1, par. 14). 55 Ibid. (1960, par. 24). 56 For a definition of the notion of analyticity as intended by Quine, cf. Definition 2.10, in Section 2.2.4. 57 Cf. Primiero (2004). 58 Already by Curry’s combinatorial logic one understands that semantical iden- tity comes not before typing, but rather the two notions are to be considered simultaneously.
- 46. 1.5. Formal Analysis of Types and Judgements 31 1.5 Formal Analysis of Types and Judgements In Section 1.5.1 the ground types, namely, the type of sets and that of propositions, will be introduced as objects definable by monomorphic types; the forms of judgements in use within the theory and their formalization will be shown. Moreover, this will also allow for the introduction of another type, that of functions and its formalization. On the basis of the theoretical analysis developed above, the formal role of the notion of identity will be explained and formal rules for it introduced. 1.5.1 Formalizing the Forms of Judgement The aim of this section is to introduce the formalization of the judgemental forms used in the type-theoretical framework. It will become clear why the notion of function and the notations explained above for this concept (those developed by Frege, Russell, and later by Church) are relevant to the notion of type considered here. I will first consider the formal expressions of the theory and try to explain how the formalization reflects the theoretical questions introduced above: conceptual priority, identity, and definition. The essential questions for defining the formal objects of the theory are: What is a type? And what does it mean for an element to be of a certain type? According to the definition of type, and the explanation of mathe- matical objects already considered in the light of Intuitionistic logic, it is not possible to know what a type is and being at the same time in doubt about the properties of the objects belonging to that type: this obviously makes the theory more trustworthy from an epistemic point of view. There are then two complementary ways to follow, in order to get the definition of “type”: (a) To know a type is to know what an object of that type is. (b) To know a type means knowing what it means for two objects of that type to be the same. Accordingly, two main forms of judgements are obtained: the first will state that there is an element belonging to a certain type (a), the second that two objects are identical within the same type (b). These expressions will together furnish the definition for the type involved. According to the con- ceptual priority explained in Section 1.3.3, the type denoted by such a definition must be meaningfully stated before, i.e. its meaningfulness is a presupposition for those judgements to be done: in the conceptual order, the type comes before its definition.59 Let us start by the formalization of type-expressions. The basic judgement α is a type 59 Cf. Martin-Löf (1993, pp. 31–32). The problem of priority between type- declarations and type-definitions is just introduced here: the classic solution pro- vided by Martin-Löf in order to avoid impredicativity is presented in Section 1.5.3. A new critical treatment of the problem and a theoretical solution is presented throughout Chapter 4. For the formal presentation of presuppositions in CTT see also Primiero [forthcoming a].
- 47. 32 1. Constructive Type Theory: Foundation and Formalization is formalized as α : type. (1.1) This is an absolute judgement, corresponding to a presupposition for any other following judgement using the type α.60 We can also state that two types are identical: α and β are identical types formally α = β : types. (1.2) In this case, we are presupposing respectively the declarations α : type and β : type . Any type declaration needs at this point to be defined, i.e. one needs to state what it means for an object a to be of the type α, i.e. to know the conditions under which one can assert a is an object of the type α, formalized as a : α. (1.3) This formula represents the application criterion, from which follows the identity criterion, the second condition in order to explain what the type α is: the latter consists of knowing when two objects of that type are the same. Obviously, when one knows what it means “being of the type α”, one already knows what it means for two objects a and b that a and b are identical objects of the type α, formalized as a = b : α. (1.4) In order to make this last assertion, one needs of course to know before and respectively that a : α, that b : α, and finally (going backwards) that α : type. Knowledge that two objects a and b are the same inside the type α also means to know if both are equal to a third object c, inside the same type α. In this way we obtain the three conditions for identity of types: – Reflexivity – Symmetry – Transitivity The rules for identical types state that: • Given two identical types, an arbitrarily given object of one of the types will also be an object of the other type: a : α α = β : type a : β (1.5) 60 In the following, when the judgement α : type works as a presupposition for another judgement (e.g. a : α), the notation α : type will be used.
- 48. 1.5. Formal Analysis of Types and Judgements 33 • Given two identical types, two identical objects of one of the types are identical objects of the other type: a = b : α α = β : type a = b : β. (1.6) This is to be satisfied for all the objects of both the types in question. The properties holding for objects belonging to types will hold for types themselves. These rules allow clearly to state definitional equality between types. 1.5.2 Formalizing Equality Rules The three identity conditions holding both for types and elements of types are formally presented in the remainder of this section.61 Equality rules for elements of a monomorphic type are: Reflexivity a : α a = a : α (1.7) Symmetry a = b : α b = a : α (1.8) Transitivity a = b : α b = c : α a = c : α. (1.9) Equality rules for types: 61 In relation to the differences between the monomorphic and the polymorphic versions of type theory, it has been mentioned at the end of Section 1.3.1 that in the switch from the latter to the former the possibility of expressing a rule of extensional equality for sets with a strong elimination rule (a “too strong” one in fact) is lost: in the original semantics of Martin-Löf (1982, 1984) judgemental equility turns out to be more general than convertibility; in Nordström, Peters- son and Smith (1990, pp. 60–61) rules of formation, introduction, and elimination for equivalence are provided, extensional with respect to substitution. The strong elimination rule used there does not express this extensionality based on struc- tural induction; therefore, it is supported by a second Eq-elimination rule. By using both, one is able to derive an induction rule corresponding to the usual Id-elimination for the semantics of the polymorphic version of CTT. In Section 1.8.1, together with examples of rules for different sets definable in terms of types, the equality sets for the monomorphic version will also be considered.
- 49. 34 1. Constructive Type Theory: Foundation and Formalization Reflexivity α : type α = α : type (1.10) Symmetry α = β : type β = α : type (1.11) Transitivity α = β : type β = γ : type α = γ : type (1.12) 1.5.3 Categories Once the formalization for judgements is introduced, together with their equality rules, the structure of the theory is completed by the defini- tion of the ground types. This leads to present the constructive notion of proposition and to explain its equivalence with the notion of set (the already mentioned Curry–Howard isomorphism). According to the Brouwer–Heyting–Kolmogorov (BHK) interpretation, instances of propo- sitions, sets, and problems are actually instances of the same concept, i.e. rules valid in one case are valid also in the others. In CTT propositions, sets and problems represent the ground types of the theory: expressions involving propositions or sets as predicates are in fact particular instances of a unique form of predication, and the same is true for the predication of an element of a set and a proof of a proposition. This means to recognize two main expression forms within the theory: . . . : type, (1.13) . . . : α, (1.14) the second expression assuming implicitly that α has been introduced as a certain type (i.e. appearing on the left side of the colon in the first kind of expression). The first of these forms introduces types, in turn amounting either to a proposition, a set, or a problem; the second introduces an object of a certain type, respectively a proof, an element, or a solution. Both the expressions are generally formulated within contexts of assumptions, of the form Γ = (x1 : α1, . . . , xn : αn).
- 50. 1.5. Formal Analysis of Types and Judgements 35 The formulation of a judgement under a context of assumptions leads to the expression of a hypothetical judgement; an empty context makes the judge- ment a categorical one.62 We can formalize the previous forms of expression as follows: type(Γ) (1.15) α(Γ) (1.16) These expressions introduce what Martin-Löf calls the categories of the theory: namely, the first introduces the category of types, the second, the category of objects of types. The word category represents a general noun for the kind of predication structures used within the theory, in the light of the Aristotelian notion of category: categories are the meaning-giving structures of the theory, in terms of types and objects belonging to them. A form of judgement is nothing but a category of reasoning (logical sense) or of knowledge (philosophical sense), and to know means to make cor- rect judgements in terms of such categories. Thus, for example, judgements falling under the first category are those like: set : type elem(N) : type, i.e. judgements stating that sets are types and that the elements of the set of natural numbers form a type. Such judgements say that something is of a certain category, they declare something to be a type. This is the proper sense in which an expression is called a type-declaration. Within the second kind of category fall those judgements declaring something to be of a certain type, e.g. N : set 0 : elem(N), i.e. judgements saying that natural numbers form a set, and that zero be- longs to the type of the elements of natural numbers. These judgements represent a derived sense by which one refers to a type-declaration, namely the declaration of the type some element belongs to. The idea of category is clearly given by abstraction from the type itself, in order to grasp those general forms of expression which are meaningful for the theory. It is exactly by introducing this notion of category as a form of expression that the problem of impredicativity for types is avoided. The relation between types and their definition has already been presented: a type is introduced by a type-declaration, such a judgement being in this way a presupposition for those judgements predicating objects within that type. The definition of the mentioned type is given exactly in terms of the latter judgements, representing the application and the identity criterion. In this formulation the notion of type itself could still be accounted as con- tradictory, in that its definition presupposes the concept, whereas only the 62 The role of context will be widely clarified in Section 1.6, where hypothetical judgements are formally introduced.
- 51. 36 1. Constructive Type Theory: Foundation and Formalization introduction of the criteria of application and identity furnishes the mean- ing of such a concept. The introduction of the notion of category is required in order to avoid such a vicious circle: the meaning of expressions is distin- guished from the meaning of the single types, in that the expressions refer to the meaning provided by the related category (as forms of expression), whereas types refer to the presence of a meaningful concept, introduced by the relevant presupposition. It is clear at this point that the concept of type escapes impredicativity by referring to the use of these meaning-giving structures, while on the other hand it is still necessary to clarify the nature of type-declaration and the definition of types in connection to the notion of meaning, namely, by explaining the nature of presuppositions.63 To un- derstand what a type is (and in turn what is one of its specifications, like set or prop), it is necessary only to grasp what an arbitrary object of that type is, i.e. one must understand which objects fall within that concept. As should be clear by now, to define the type set or prop, one needs to know respectively how canonical elements of a set can be formed, or how to show an effective construction for a proposition.64 Once the categories are introduced, the notion of type is a primitive concept, introduced by the first general form of judgement (α is a type — formula 1.1). Such a judgement resumes thus all the different possible interpretations: it can in fact be read in different ways, after one states what the ground types are. In particular, we can give the following expressions as valid examples of the first form of category: prop : type set : type prob : type stating respectively the ground types of sets, propositions, and problems. They are all equivalent forms, coming from the definition of Intuitionistic logic, of constructive set theory, and from the reading of Kolmogorov (1932), according to which a problem is identified with the set of its solutions (the already mentioned BHK interpretation). On this basis, CTT was designed as a logic for mathematical reasoning, which through the computational content of constructive proofs can be used as a programming logic.65 The forms of judgements A : set A : prop are in fact different versions of the same form of expression, because a set is defined by explaining how its canonical elements are formed, while a proposition is defined by laying down the set of its proofs.66 63 This analysis is done in Section 3.2.1, and more extensively in Primiero (forth- coming a). 64 Martin-Löf (1984, p. 22). 65 For a development of this theory in terms of a programming language, cf. Nordström, Petersson and Smith (1990). 66 In the following the formalization of the ground types and their rules will be presented. To this aim, remarks about the notation are needed: the type of
- 52. 1.5. Formal Analysis of Types and Judgements 37 1.5.4 Type set Sets are thus introduced as a ground type (set : type), and a certain set A (A : set) is known if one knows how to form canonical elements for this set (a ∈ A), and when two of its canonical elements are equal (a = b ∈ A), which represents the canonical definition of a type. Moreover, two sets are equal if a canonical element of one set is always a canonical element of the other set, and if two elements which are equal inside one of these sets are equal inside the other as well (equality for types). The notion of set has different possible interpretations: – Class theory (where “class” is some subset of the universe of discourse) – Cantor’s set theory (where “set” is an intuitive description of the universe of discourse)67 – Formalized set theory (where “set” is an iterative or cumulative notion)68 In the type-theoretical framework proposed by Martin-Löf, the notion of set is defined according to a combination of logic and set theory, in which “set” is distinct both from class and iterative hierarchy, using instead the defining criteria. The rules stating that set is a type and that it is the type of a certain A, are the following: set : type set = set : type (1.18) A : set A : type A = B : set A = B : type (1.19) A being a set, the elements of A define a type: A : set El(A) : type A = B : set El(A) = El(B) : type (1.20) That a is an element of the set A is formally expressed both by a : El(A) (1.21) sets and that of propositions (set and prop) will always be represented by capital letters (second form of category); Greek letters will be used only for monomorphic types (first form of category); whereas the symbol ∈ refers to set-theorethical expressions, in general the use of the colon a : A is preferred, holding both for elements of sets and for proofs of propositions. Finally, the more common symbol ∀ instead of the proper Π is used also for sets, and this will be in common with the rule for the type prop, via the following definitional equality: (∀x ∈ A)B(x) =def (Πx ∈ A)B(x) (1.17) Cf. Martin-Löf (1984, p. 32). 67 Cantor (1878). 68 Set theory has in fact also a type-theoretic interpretation and a related con- structive version, introduced by Myhill (1975) and further explored, for example, by Aczel (1978, 1982, 1986) and Aczel and Rathjen (2001).
- 53. 38 1. Constructive Type Theory: Foundation and Formalization a ∈ A (1.22) To make an example of a set definable in terms of types, let us consider the set of natural numbers: one will need to make a judgement declaring such a collection of elements to be of a certain type, namely of the type set. The axioms used are exactly the type-theoretical counterpart of the first two Peano axioms, plus the type-declaration of N being a set: N : set; 0 : N; a : N s(a) : N These are the formal rules for canonical elements of this set. By the identity criterion, we need to know when two elements of such a set are equal, starting from zero and using the successor rule: 0 = 0 : N; a = b : N s(a) = s(b) : N This represents a method which when executed yields a canonical element of the set as result, and correspondingly two elements are equal if the re- spective methods yield equal canonical elements. In Section 1.8 the compu- tational rules for types will be formally and explicitly introduced, and some other examples will be provided for sets definable in terms of the monomor- phic type theory. Once the type of sets is introduced, more attention can be given to the type of propositions. 1.5.5 Type prop The ground type of propositions, prop, is explained by laying down the axiom prop : type (1.23) and furnishing a justification for the following judgement: A is a proposition, A : prop This judgement is explained by answering two questions: “what is a propo- sition?”, which represents the application criterion, and “what does it mean for two propositions to be the same?”, which corresponds to the identity criterion. In CTT the first question requires an epistemological analysis, which relies on the more general philosophical question: “what is it to know a proposition?” The classical solution and interpretation of the notion of proposition, given by Aristotle, is that “a proposition (ἀπόφανσις) is what can be true or false”, and to know which is the case one has to know the state of affairs (ontology) to which the proposition refers, so that it is not the case that “the snow is white” is true because we affirm it, rather the other way round, i.e. the proposition is true if the snow actually happens to be white. In the history of modern logic, this has been notoriously translated
- 54. 1.5. Formal Analysis of Types and Judgements 39 by Boole as “a proposition is what has a truth-value, 1 or 0”,69 and this has been formally developed by the truth tables for connectives. Frege de- fined the concept of proposition in his Grundgesetze der Arithmetik on the basis of the “truth-conditions” for logical operators, developing later such conditions by considering the role of Bedeutung. The now common “truth- tables”, introduced by Wittgenstein in his Tractatus Logico-Philosophicus and later also by Post and Lukasiewicz, can be summarized as follows: Explanation of propositions in terms of truth-conditions A true B true A ∧ B true A true A ∨ B true B true A ∨ B true (A true) B true A ⊃ B ⊥: false (x ∈ D) P(x) true (∀x ∈ D)P(x) true d ∈ D P(d) true (∃x ∈ D)P(x) true As known, by means of these tables it is possible to formalize the laws of classical logic with quantification over a finite domain; difficulties arise in the Boolean interpretation when one needs quantified propositions over infinite domains, such as in the following two laws: A(x) : prop (∀x)A(x) : prop A(x) : prop (∃x)A(x) : prop (1.24) Moreover, in what we can refer to as the “Fregean–Wittgensteinean inter- pretation” of the notion of proposition another well-known problem arises: if only truth-conditions are needed in order to define a proposition, obvi- ously all truths (such as all falsities) are identical propositions, because the principle of identity is based upon the truth-conditions, so that identity cor- responds to material equivalence. On the basis of these remarks a general philosophical critique of classical logic, essentially regarding the role of the law of excluded middle, was developed by the Intuitionists, thus producing a new interpretation of the notion of proposition. 69 This means that if we previously define a set like Bool by the domain {1, 0}, we can then define a proposition just as an element of that set.
- 55. Exploring the Variety of Random Documents with Different Content
- 59. The Project Gutenberg eBook of Zones of the Spirit: A Book of Thoughts
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Zones of the Spirit: A Book of Thoughts Author: August Strindberg Commentator: Arthur Babillotte Translator: Claud Field Release date: November 6, 2013 [eBook #44118] Most recently updated: October 23, 2024 Language: English Credits: Produced by Marc D'Hooghe (Images generously made available by the Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK ZONES OF THE SPIRIT: A BOOK OF THOUGHTS ***
- 61. ZONES OF THE SPIRIT A BOOK OF THOUGHTS BY
- 62. AUGUST STRINDBERG AUTHOR OF THE INFERNO, THE SON OF A SERVANT, ETC. WITH AN INTRODUCTION BY ARTHUR BABILLOTTE TRANSLATED BY CLAUD FIELD, M.A. G.P. PUTNAM'S SONS NEW YORK AND LONDON The Knickerbocker press 1913 INTRODUCTION Seldom has a man gone through such profound religious changes as this Swede, who died last May. The demonic element in him, which spurred him on restlessly, made him scale heaven and fathom hell, gave him glimpses of bliss and damnation. He bore the Cain's mark on his brow: A fugitive and a wanderer shalt thou be.
- 63. He was fundamentally religious, for everyone who searches after God is so,—a commonplace truth certainly, but one which needs to be constantly reiterated. And Strindberg's search was more painful, exact, and persevering than that of most people. He was never content with superficial formulas, but pressed to the heart of the matter, and followed each winding of the labyrinthine problem with endless patience. Too often the Divinity which he thought he had discovered turned out a delusion, to be scornfully rejected the moment afterwards. Until he found the God, whom he worshipped to the end of his days, and whose existence he resolutely maintained against deniers. As a child he had been brought up in devout belief in God, in submission to the injustice of life, and in faith in a better hereafter. He regarded God as a Father, to Whom he made known his little wants and anxieties. But a youth with hard experiences followed his childhood. The struggle for daily bread began, and his heavenly Father seemed to fail him. He appeared to regard unmoved, from some Olympian height, the desperate struggles of humanity below. Then the defiant element which slumbered in Strindberg wrathfully awoke, and he gradually developed into a free-thinker. It fared with him as it often does with young and independent characters who think. Beginning with dissent from this and that ecclesiastical dogma, his criticism embraced an ever-widening range, and became keener and more unsparing. At last every barrier of respect and reverence fell, the defiant spirit of youth broke like a flood over all religious dogmas, swept them away, and did not stop short of criticising God Himself. Meanwhile his daily life, with its hard experiences, went on. Books written from every conceivable point of view came into his hands. Greedy for knowledge as he was, he read them all. Those of the free-thinkers supported his freshly aroused incredulity, which as yet needed support. His study of philosophical and scientific works made a clean sweep of what relics of faith remained. Anxiety about his daily bread, attacks from all sides, the alienation of his friends, all contributed towards making the free-thinker into an atheist. How
- 64. can there be a God when the world is so full of ugliness, of deceit, of dishonour, of vulgarity? This question was bound to be raised at last. About this time he wrote the New Kingdom, full of sharp criticisms of society and Christianity. As an atheist Strindberg made various attempts to come to terms with the existing state of things. But being a genius out of harmony with his contemporaries, and always longing for some vaster, fairer future, this was impossible for him. When he found that he came to no goal, a perpetual unrest tortured him. His earlier autobiographic writings appeared, marked by a strong misanthropy, and composed with an obscure consciousness of the curse: A fugitive and a wanderer shalt thou be. At last his consciousness becomes clear and defined. He recognises that he is a lost soul in hell already, though outwardly on earth. This was the most extraordinary period in Strindberg's life. He lived in the Quartier Latin in Paris, in a barely furnished room, with retorts and chemical apparatus, like a second Faust at the end of the nineteenth century. By experiments he discovered the presence of carbon in sulphur, and considered that by doing so he had solved a great problem, upset the ruling systems of chemistry, and gained for himself the only immortality allowed to mortals. He came to the conclusion that the reason why he had gradually become an atheist was that the Unknown Powers had left the world so long without a sign of themselves. The discovery made him thankful, and he lamented that he had no one to thank. From that time the belief in unknown powers grew stronger and stronger in him. It seems to have been the result of an almost complete, long, and painful solitude. At this time his brain worked more feverishly, and his nerves were more sensitive than usual. At last he reached the (for an atheist) astounding conclusion: When I think over my lot, I recognise that invisible Hand which disciplines and chastens me, without my knowing its purpose. Must I be humbled in order to be lifted up, lowered in order to be raised? The thought continually recurs to me,
- 65. 'Providence is planning something with thee, and this is the beginning of thy education.'[1] Soon after this he gave up his chemical experiments and took up alchemy, with a conviction, almost pathetic in its intensity, that he would succeed in making gold. Although his dramas had already been performed in Paris, a success which had fallen to the lot of no other Swedish dramatist, he forgot all his successes as an author, and devoted himself solely to this new pursuit, to meet again with disappointment. On March 29, 1897, he began the study of Swedenborg, the Northern Seer. A feeling of home-sickness after heaven laid hold of him, and he began to believe that he was being prepared for a higher existence. I despise the earth, he writes, this unclean world, these men and their works. I seem to myself a righteous man, like Job, whom the Eternal is putting to the test, and whom the purgatorial fires of this world will soon make worthy of a speedy deliverance. More and more he seemed to approach Catholicism. One day he, the former socialist and atheist, bought a rosary. It is pretty, he said, and the evil spirits fear the cross. At the same time, it must be confessed that this transition to the Christian point of view did not subdue his egotism and independence of character. It is my duty, he said, to fight for the maintenance of my ego against all influences which a sect or party, from love of proselytising, might bring to bear upon it. The conscience, which the grace of my Divine protector has given me, tells me that. And then comes a sentence full of joy and sorrow alike, which seems to obliterate his whole past. Born with a home-sick longing after heaven, as a child I wept over the squalor of existence and felt myself strange and homeless among men. From childhood upwards I have looked for God and found the Devil. He becomes actually humble, and recognises that God, on account of his pride, his conceit, his ὕβρις, had sent him for a time to hell. Happy is he whom God punishes.
- 66. The return to Christ is complete. All his faith, all his hope now rest solely on the Crucified, whom he had once demoniacally hated. He now devoted himself entirely to the study of Swedenborg. He felt that in some way the life of this strange man had foreshadowed his own. Just as Swedenborg (1688-1772) had passed from the profession of a mathematician to that of a theologian, a mystic, and finally a ghost-seer and theosoph, so Strindberg passed from the worldly calling of a romance-writer to that of a preacher of Christian patience and reconciliation. He had occasional relapses into his old perverse moods, but the attacks of the rebellious spirit were weaker and weaker. He told a friend who asked his opinion regarding the theosophical concept of Karma, that it was impossible for him to belong to a party which denied a personal God, Who alone could satisfy his religious needs. In a life so full of intellectual activity as his had been, Strindberg had amassed an enormous amount of miscellaneous knowledge. When he was nearly sixty he began to collect and arrange all his experiences and investigations from the point of view he had then attained. Thus was composed his last important work, Das Blau Buch, a book of amazing copiousness and originality. Regarding it, the Norwegian author Nils Kjaer writes in the periodical Verdens Gang: More comprehensive than any modern collection of aphorisms, chaotic as the Koran, wrathful as Isaiah, as full of occult things as the Bible, more entertaining than any romance, keener-edged than most pamphlets, mystical as the Cabbala, subtle as the scholastic theology, sincere as Rousseau's confession, stamped with the impress of incomparable originality, every sentence shining like luminous letters in the darkness—such is this book in which the remarkable writer makes a final reckoning with his time and proclaims his faith, as pugnaciously as though he were a descendant of the hero of Lutzen. The book, in truth, forms a world apart, from which all lying, hypocrisy, and conventional contentment is banished; in it is heard the stormy laughter of a genius who has freed himself from the fetters of earth, the proclamation of the creed of a strange Christian who interprets and reveres Christ in his own fashion, the challenge of an original and
- 67. creative mind which believes in its own continuance, the expression of the yearning of a lonely soul to place itself in harmonious relations with the universe. An especially interesting feature of the Blau Buch is the expression of Strindberg's views regarding the great poets, artists, and thinkers of the past and present. He speaks of Wagner and Nietzsche, the two antipodes; of Horace, who, after many wanderings, recognised the hand of God; of Shakespeare, who had lived through the experience of every character he created; of Goethe, regarding whom he remarks, with evident satisfaction, In old age, when he grew wise, he became a mystic, i.e. he recognised that there are things in heaven and earth of which the Philistines never dream. Of Maeterlinck, he says, He knows how to caricature his own fairest creations; and accuses Oscar Wilde of want of originality. Regarding Hegel, he notes with pleasure that at the end of his life he returned to Christianity. With deep satisfaction he writes, Hegel, after having gone very roundabout ways, died in 1831, of cholera, as a simple, believing Christian, putting aside all philosophy and praying penitential psalms. In Rousseau he recognises a kindred spirit, in so far as the Frenchman, like himself, hated all that was unnatural. One can agree with Rousseau when he says, 'All that comes from the Creator's hand is perfect, but when it falls into the hands of man it is spoilt.' The Blau Buch marks the summit of Strindberg's chequered sixty years' pilgrimage. Beneath him lies the varicoloured landscape of his past life, now lit up with gleams of sunshine, now draped in dark mists, now drowned in storms of rain. But Strindberg, the poet and thinker, has escaped from both dark and bright days alike; he stands peacefully on the summit, above the trivialities, the cares, and bitternesses of life, a free man. He is like Prometheus, fettered to the rock for having bestowed on men the gift of fire, but liberated after he has learnt his lesson. In his calm is something resembling the dignity of Goethe's old age. As the latter sat on the Kickelhahn, looking down on Thuringia, and saw the panorama of his life pass before him, so Strindberg takes a retrospect in his Blau Buch. It is
- 68. the canticle of his life, a hymn of thankfulness for the recovered faith in which he has found peace. At its conclusion he thus sums up: Rousseau's early doctrine regarding the curse of mere learning should be repondered. A new Descartes should arise and teach the men to doubt the untruths of the sciences. Another Kant should write a new Critique of Pure Reason and re- establish the doctrine of the Categorical Imperative, which, however, is already to be found in the Ten Commandments and the Gospels. A prophet should be born to teach men the simple meaning of life in a few words. It has already been so well summed up: 'Fear God, and keep His commandments,' or 'Pray and work.' All the errors and mistakes which we have made should serve to instil into us a lively hatred of evil, and to impart a fresh impulse to good; these we can take with us to the other side, where they will bloom and bear fruit. That is the true meaning of life, at which the obstinate and impenitent cavil, in order to save themselves trouble. Pray, but work; suffer, but hope; keeping both the earth and the stars in view. Do not try and settle permanently, for it is a place of pilgrimage; not a home, but a halting-place. Seek the truth, for it is to be found, but only in one place, with the One who Himself is the Way, the Truth, and the Life. ARTHUR BABILLOTTE. [1] Strindberg's Inferno. CONTENTS THE HISTORY OF THE BLUE BOOK A BLUE BOOK—
- 69. The Thirteenth Axiom The Rustic Intelligence of the Beans The Hoopoo, or An Unusual Occurrence Bad Digestion The Song of the Sawyers Al Mansur in the Gymnasium The Nightingale in the Vineyard The Miracle of the Corn-crakes Corollaries Phantasms which are Real Crex, Crex! The Electric Battery and the Earth Circuit Improper and Unanswerable Questions Superstition and Non-Superstition Through Faith to Knowledge The Enchanted Room Concerning Correspondences The Green Island Swedenborg's Hell Preliminary Knowledge Necessary Perverse Science Truth in Error Accumulators Eternal Punishment Desolation A World of Delusion The Conversion of the Cheerful Pagan, Horace Cheerful Paganism and its Doctrine of Hell Faith the Chief Thing Penitents Paying for Others The Lice-King The Art of Life The Mitigation of Destiny The Good and the Evil
- 70. Modesty and the Sense of Justice Derelicts Human Fate Dark Rays Blind and Deaf The Disrobing Chamber The Character Mask Youth and Folly When I was Young and Stupid Constant Illusions The Merits of the Multiplication-Table Under the Prince of this World The Idea of Hell Self-Knowledge Somnambulism and Clairvoyance in Everyday Life Practical Measures against Enemies The Goddess of Reason Stars Seen by Daylight The Right to Remorse A Religious Theatre Through Constraint to Freedom The Praise of Folly The Inevitable The Poet's Sacrifice The Function of the Philistines World-Religion The Return of Christ Correspondences Good Words Severe and not Severe Yeast and Bread The Man of Development Sins of Thought Sins of Will The Study of Mankind
- 71. Friend Zero Affable Men Cringing before the Beast Ecclesia Triumphans Logic in Neurasthenia My Caricature The Inexplicable Old-time Religion The Seduced become Seducers Large-hearted Christianity Reconnection with the Aërial Wire The Art of Conversion The Superman To be a Christian is not to be a Pietist Strength and Value of Words The Black Illuminati Anthropomorphism Fury-worship as a Penal Hallucination Amerigo or Columbus A Circumnavigator of the Globe The Poet's Children Faithful in Little Things The Unpracticalness of Husk-eating A Youthful Dream for Seven Shillings Envy Nobody! The Galley-slaves of Ambition Hard to Disentangle The Art of Settling Accounts Growing Old Gracefully The Eight Wild Beasts Deaf and Blind Recollections Children are Wonder-Children Men-resembling Men Christ is Risen Revolution-Sheep
- 72. Life Woven of the Same Stuff as our Dreams The Gospel of the Pagans Punished by the Imagination Bankruptcy of Philosophy A Whole Life in an Hour The After-Odour Peaches and Turnips The Web of Lies Lethe A Suffering God The Atonement When Nations Go Mad The Poison of Lies Murderous Lies Innocent Guilt The Charm of Old Age The Ring-System Lust, Hate, and Fear, or the Religion of the Heathen Whom the Gods Wish to Destroy The Slavery of the Prophet Absurd Problems The Crooked Rib White Slavery Noodles Inextricable Confusion Phantoms Mirage Pictures Trifle not with Love A Taking Religion The Sixth Sense Exteriorisation of Sensibility Telepathic Perception Morse Telepathy Nisus Formativus, or Unconscious Sculpture
- 73. Projections Apparitions The Reactionary Type The Hate of Parasites A Letter from the Dead A Letter from Hell An Unconscious Medium The Revenant The Meeting in the Convent Correspondences Portents The Difficult Art of Lying Religion and Scientific Intuition The Freed Thinker Primus inter pares Heathen Imaginations Thought Bound by Law Credo quia (et-si) absurdum The Fear of Heaven The Goat-god Pan and the Fear of the Pan- pipe Their Gospel The Deposition of the Apes The Secret of the Cross Examination and Summer Holidays Veering and Tacking Attraction and Repulsion The Double Paw or Hand The Thousand-Years' Night of the Apes The Favourite Scientific Villainies Necrobiosis, i.e. Death and Resurrection Secret Judgment Hammurabi's Inspired Laws Received from the Sun-God
- 74. Strauss's Life of Christ Christianity and Radicalism Where are We? Hegel's Christianity Men of God's Hand Night-Owls Apotheosis Painting Things Black The Thorn in the Flesh Despair and Grace The Last Act Consequences of Learning Rousseau Rousseau Again Materialised Apparitions The Art of Dying Can Philosophy Bring any Blessing to Mankind? Goethe on the Bible Now we Can Fly Too! Hurrah The Fall and Original Sin The Gospel Religious Heathen The Pleasure-Garden The Happiness of Love Our Best Feelings Blood-Fraternity The Power of Love The Box on the Ear Saul, afterwards Called Paul A Scene from Hell The Jewel-Casket or his Better Half The Mummy-Coffin In the Attic The Sculptor On the Threshold at Five Years of Age
- 75. Goethe on Christianity and Science Summa Summarum Zones of the Spirit THE HISTORY OF THE BLUE BOOK (Prefixed to the Third Swedish Edition) I had read how Goethe had once intended to write a Breviarium Universale, a book of edification for the adherents of all religions. In my Historical Miniatures I have attempted to trace God's ways in the history of the world; I included Christianity in my survey by commencing with Israel, but perhaps I made the mistake of ranging other religions by the side of Christianity, while they ought to have stood below it. A year passed. I felt myself constrained by inward impulses to write a fairly unsectarian breviary; a word of wisdom for each day in the year. For that purpose I collected the sacred books of all religions, in order to extract from them sayings on which to write. But the books did not open themselves to me! The Vedas and Zend-Avesta were sealed, and did not yield a single saying; only the Koran gave one, but that was a lion! (page 45). Then I determined to alter my design. I formed the plan of writing apothegms of simply worldly wisdom regarding men, and of calling the book Herbarium Humane. But I postponed the work since I trembled at the greatness of the task and the crudity of my plan. Then came June 15, 1906. As I took my morning walk, the first thing I saw was a tramcar with the
- 76. number 365. I was struck by this number, and thought of the 365 pages which I intended to write. As I went on, I entered a narrow street. A cart went along by my side carrying a red flag; it was a powder-flag. The cart kept parallel with me and began to disturb me. In order to escape the sight of the powder-flag, I looked up in the air, and there an enormous red flag (the English one) flaunted conspicuously before my eyes. I looked down again, and a lady dressed in black, with a fiery-red hat, was crossing the street in a slanting direction. I hastened my steps. Immediately my eyes fell on the window of a stationer's shop; in it a piece of cardboard was displayed, bearing the word Herbarium. It was natural that all this should make an impression on me. My resolution was now taken; I laid down the plan of my powder- chamber, which was to become the Blue Book. A year passed, slowly, painfully. The most remarkable thing that happened was this. They began to rehearse my drama, the Dream Play, in the theatre; simultaneously, a change took place in my daily life. My servant left me; my domestic arrangements were upset; within forty days I had six changes of servants—one worse than the other. At last I had to serve myself, lay the table and light the stove. I ate black broken victuals out of a basket. In short, I had to taste the whole bitterness of life without knowing why. One morning during this fasting period I passed by a shop window in which I saw a piece of tapestry which attracted and delighted me. I thought I saw my dream-play in the design woven on the tapestry. Above was the growing castle, and underneath the green island over-arched by a rainbow, and with Alpine summits illumined by the sun. Round it was the sea reflecting the stars and a great green sea- snake partly visible; low down in the border was a row of fylfots— the symbol Swastika, signifying good-luck. That was, at any rate, my meaning; the artist had intended something else which does not belong here.
- 77. Then came the dress-rehearsal of the Dream Play. This drama I wrote seven years ago, after a period of forty days' suffering which were among the worst which I had ever undergone. And now again exactly forty days of fasting and pain had passed. There seems, therefore, to be a secret legislature which promulgates clearly defined sentences. I thought of the forty days of the flood, the forty years of wandering in the desert, the forty days' fast kept by Moses, Elijah, and Christ. My journal thus records my impressions: The sun shines. A certain quiet resigned uncertainty reigns within me. I ask myself whether a catastrophe will not prevent the performance of the piece, which perhaps ought not to be played. In it I have, at any rate, spoken men fair, but to advise the Ruler of the Universe is presumption, perhaps blasphemy. The fact that I have laid bare the comparative nothingness of life (with Buddhism), its irrational contradictions, its wickedness and lawlessness, may be praiseworthy if it teaches men resignation. That I have shown the comparative innocence of men in this life, which of itself involves guilt, is not indeed wrong, but.... Just now comes a telephone message from the theatre: The result of this is in God's hand. Exactly what I think, I answer, and ask myself again whether the piece ought to be played. (I believe it is already determined by the higher powers what the issue of the first performance will prove.) I feel as though it were Sunday. The White Shape appears outside on the balcony of the growing castle. My thoughts have lately been occupied with death and with the life after this. Yesterday I read Plato's Timæus and Phædo. At present I write a work called The Island of the Dead. In it I describe the awakening after death, and what follows. But I hesitate, for I am frightened at the boundless misery of mere life. Lately I burned a drama; it was so sincere, that I shuddered at it. What I do not understand is this: ought one to hide the misery, and flatter men? I
- 78. wish to write cheerfully and beautifully, but ought not, and cannot. I conceive it as a terrible duty to be truthful, and life is indescribably hideous. Now the clock strikes eleven, and at twelve o'clock is the rehearsal. The same day at 8 P.M. I have seen the rehearsal of the Dream Play, and suffered greatly. I received the impression that this piece ought not to be played. It is presumptuous, and certainly blasphemous (?). I am disturbed and alarmed. I have had no midday meal; at seven o'clock I ate some cold food out of the basket in the kitchen. During the religious broodings of my last forty days I read the Book of Job, saying to myself certainly at the same time that I was no righteous man like him. Then I came to the 22nd chapter, in which Eliphaz the Temanite unmasks Job: Thou hast taken pledges of thy brother for nought, and stripped the naked of their clothing; thou hast not given water to the weary to drink, and thou hast withholden bread from the hungry. ... Is not thy wickedness great and thine iniquities infinite? Then the whole comfort of the Book of Job vanished, and I stood again forlorn and irresolute. What shall a poor man hold on to? What shall I believe? How can he help thinking perversely? Yesterday I read Plato's Timæus and Phædo. There I found so much self-contradictory wisdom, that in the evening I threw my devotional books away and prayed to God out of a full heart. What will happen now? God help me! Amen. The stage-manager visited me yesterday evening. We both felt, in despair.... The night was quiet. April 16, 1907.—Read the proof of the Black Flags,[1] which I wrote in 1904. I asked myself whether the book was a crime, and whether it ought to be published. I opened the Bible, and came on the prophet Jonah, who was compelled to prophesy although he hid himself. That quieted me. But it is a terrible book!
- 79. April 17.—To-day the Dream Play will be performed for the first time. A gentle fall of snow in the morning. Read the last chapter of Job: God punishes Job because he presumed to wish to understand His work. Job prays for pardon, and is forgiven. Quiet grey weather till 3 P.M. Then G. came with a piece of good news. Spent the evening alone at home. At eight o'clock there was a ring at the door. A messenger brought a laurel-wreath with the inscription: Truth, Light, Liberation. I took the wreath at once to the bust of Beethoven on the tiled stove and placed it on his head, since I had so much to thank him for, especially just now for the music accompanying my drama. At eleven o'clock a telephone from the theatre announces that everything has gone well. May 29.—The Black Flags come out to-day. I make very satisfactory terms with the publisher regarding the Blue Book (and I had thought it would not be printed at all). So I determined to remain in my house, which I had determined to leave on account of poverty. August 20.—I read this evening the proofs of the Blue Book. Then the sky grew coal-black with towering dark clouds. A storm of rain fell; then it cleared up, and a great rainbow stood round the church, which was lit up by the sun. August 22.—I am reading now the proofs of the Blue Book, and I feel now as though my mission in life were ended. I have been able to say all I had to say. I dreamt that I was in the home of my childhood at Sabbatsberg, and saw that the great pond was dried up. This pond had always been dangerous to children because it was surrounded by a swamp; it had an evil smell, and was full of frogs, hedgehogs, and lizards. Now in my dream I walked about on the dry ground, and was astonished to find it so clean. I thought now that I have broken with the Black Flags the frog-swamp is done with.
- 80. September 1.—Read the last proofs of the Blue Book. September 2.—Came across tramcar 365, which I had not seen since I began to write the Blue Book on June 15, 1906. September 12.—The Blue Book appears to-day. It is the first clear day in summer. I dreamt I found myself in a stone-quarry, and could neither go up nor down. I thought quite quietly, Well, I must cry for help! The German motto to-day on the tear-off calendar is: What is to be clarified must first ferment. To-day I got new clothes which fitted. My old ones had been too tight to the point of torture. My little daughter visited me. I took her home again in a chaise. September 14.—The whole day clear. Towards evening, however, about a quarter to six, the sky became covered with most portentous-looking clouds, with black outlines like obliquely hanging theatre-flies. Afterwards these were driven out by a storm over the sea. This evening my Crown Bride was performed. Thus, then, the Blue Book had appeared. It looked well with its blue and red binding, which resembled that of my first book, the Red Room, but in its contents differed as much from it as red from blue. In the first I had, like Jeremiah, to pluck up, break down, and destroy; but in this book I was able to build and to plant. And I will conclude with Hezekiah's song of praise: I said, in the noontide of my days, I shall go to the gates of the grave: My age is departed, and is removed from me as a shepherd's tent: I have rolled up like a weaver my life; he will cut me off from the loom. From day even to night wilt thou make an end of me.
- 81. Like a swallow or a crane, so did I chatter; I did mourn as a dove: mine eyes fail with looking upward. Lord, I am oppressed; undertake for me. What shall I say? He hath both spoken unto me, and himself hath done it. Behold, it was for my peace that I had great bitterness; Thou hast in love to my soul delivered it from the pit of corruption. The living, the living, he shall praise thee, as I do this day. The father to the children shall make known thy truth. I saw beforehand what awaited me if I broke with the Black Flags. But I placed my soul in God's hands, and went forwards. I affix as a motto to the following book, He who departeth from evil, maketh himself a prey. The strangest thing, however, is that from this moment my own Karma began to complete itself. I was protected, things went well with me, I found better friends than those I had lost. Now I am inclined to ascribe all my former mischances to the fact that I served the Black Flags. There was no blessing with them! [1] A roman à clef in which Strindberg fiercely attacks the Bohemians and emancipated women of Stockholm. A BLUE BOOK The Thirteenth Axiom.—Euclid's twelfth axiom, as is well known, runs thus: When one straight line cuts two other straight lines so
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